RF1:=recformat<mat:SeqEnum,dtb:SeqEnum>;
 list:=

[ PowerStructure(Rec) |
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 2, 0, 2, 0, 0, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 2, 0, 0, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 2, 2, 2, 2, 2, 2, 0, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 2, 2, 2, 0, 2, 2, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 2, 0, 0, 2, 2, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 3, 2, 2, 2, 2, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 2, 1, 0, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 0, 2, 3, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 2, 2, 1, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 2, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 2, 0, 2, 0, 0, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57792>, <20, 1020672>, <24, 8043952>, <28, 39275776>, 
            <32, 143400188>, <36, 302662656>, <40, 548362384>, <44, 746519552>, <48, 785739312>, <52, 716001792>, <56, 
            493567968>, <60, 288353792>, <64, 142519622>, <68, 49480704>, <72, 22590496>, <76, 3980288>, <80, 2887392>, 
            <84, 185088>, <88, 291184>, <92, 3328>, <96, 22076>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 0, 0, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 0, 2, 0, 0, 2, 2, 0, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 3, 2, 2, 2, 0, 2, 2, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 2, 2, 2, 2, 0, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 1, 2, 2, 2, 2, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 1, 0, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 0, 2, 1, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 2, 1, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2, 1, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 2, 2, 0, 2, 2, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 59872>, <20, 1010432>, <24, 8043792>, <28, 39261440>, 
            <32, 143467612>, <36, 302851072>, <40, 547980720>, <44, 746134528>, <48, 786492784>, <52, 716317184>, <56, 
            492913568>, <60, 288251392>, <64, 142711302>, <68, 49505280>, <72, 22644832>, <76, 3955712>, <80, 2855296>, 
            <84, 191232>, <88, 289872>, <92, 5376>, <96, 22556>, <104, 1264>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 2, 0, 2, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 2, 2, 2, 0, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 0, 0, 0, 2, 0, 2, 2, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 2, 2, 2, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 3, 2, 0, 0, 2, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 1, 2, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 2, 0, 3, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 2, 2, 1, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 2, 2, 2, 1, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 2, 2, 0, 2, 0, 0, 0, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 2, 2, 0, 2, 2, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 2, 0, 2, 2, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58496>, <20, 1009920>, <24, 8081968>, <28, 39261952>, 
            <32, 143253692>, <36, 302861312>, <40, 548527760>, <44, 746091520>, <48, 785735600>, <52, 716396032>, <56, 
            493484768>, <60, 288172544>, <64, 142520006>, <68, 49548288>, <72, 22632992>, <76, 3945472>, <80, 2874016>, 
            <84, 190720>, <88, 292336>, <92, 5888>, <96, 20732>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 0, 0, 0, 0, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 0, 2, 0, 2, 2, 2, 2, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 2, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 2, 2, 0, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 1, 2, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 0, 0, 1, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 2, 2, 2, 3, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 3, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 2, 0, 2, 0, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 2, 2, 2, 2, 2, 0, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58560>, <20, 1011712>, <24, 8073136>, <28, 39260160>, 
            <32, 143308796>, <36, 302825472>, <40, 548382864>, <44, 746242048>, <48, 785935408>, <52, 716120064>, <56, 
            493339616>, <60, 288448512>, <64, 142562118>, <68, 49397760>, <72, 22639648>, <76, 3981312>, <80, 2870752>, 
            <84, 192512>, <88, 289648>, <92, 4096>, <96, 21820>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 0, 0, 2, 0, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 3, 0, 0, 0, 2, 2, 0, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 3, 2, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 2, 0, 3, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 2, 0, 3, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 2, 2, 0, 2, 2, 2, 0, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 2, 2, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57504>, <20, 1028864>, <24, 7990672>, <28, 39292160>, 
            <32, 143634460>, <36, 302515200>, <40, 547974064>, <44, 746732544>, <48, 786040560>, <52, 716001792>, <56, 
            493450912>, <60, 288140800>, <64, 142551942>, <68, 49628160>, <72, 22589024>, <76, 3963904>, <80, 2875712>, 
            <84, 176896>, <88, 294096>, <92, 3328>, <96, 23260>, <104, 1264>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 2, 2, 2, 0, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 3, 2, 2, 0, 2, 2, 2, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 3, 0, 0, 0, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 3, 0, 0, 2, 0, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 0, 3, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 2, 0, 1, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 1, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 2, 2, 2, 0, 0, 0, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57248>, <20, 1028096>, <24, 7996304>, <28, 39292928>, 
            <32, 143604508>, <36, 302530560>, <40, 548047792>, <44, 746668032>, <48, 785943792>, <52, 716120064>, <56, 
            493515424>, <60, 288022528>, <64, 142541190>, <68, 49692672>, <72, 22576736>, <76, 3948544>, <80, 2882624>, 
            <84, 176128>, <88, 293584>, <92, 4096>, <96, 23004>, <104, 1264>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 2, 2, 2, 0, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 0, 2, 2, 0, 2, 0, 2, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 2, 0, 2, 0, 2, 0, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 2, 3, 2, 0, 0, 2, 2, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 1, 0, 0, 2, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 1, 0, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 2, 3, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 2, 3, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 0, 2, 2, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 2, 0, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57296>, <20, 1028608>, <24, 8000896>, <28, 39292416>, 
            <32, 143573356>, <36, 302520320>, <40, 548134080>, <44, 746711040>, <48, 785816272>, <52, 716041216>, <56, 
            493622272>, <60, 288101376>, <64, 142491686>, <68, 49649664>, <72, 22589568>, <76, 3958784>, <80, 2878256>, 
            <84, 176640>, <88, 296064>, <92, 3584>, <96, 22508>, <104, 1216>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 2, 0, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 0, 0, 2, 0, 2, 2, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 3, 2, 2, 0, 2, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 3, 0, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 2, 1, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 0, 0, 3, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 2, 2, 0, 2, 0, 0, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58464>, <20, 1012224>, <24, 8076112>, <28, 39259648>, 
            <32, 143292508>, <36, 302815232>, <40, 548421616>, <44, 746285056>, <48, 785890416>, <52, 716041216>, <56, 
            493357088>, <60, 288527360>, <64, 142578694>, <68, 49354752>, <72, 22615776>, <76, 3991552>, <80, 2882560>, 
            <84, 193024>, <88, 287120>, <92, 3584>, <96, 22044>, <104, 1072>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 0, 2, 2, 0, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 3, 0, 2, 2, 2, 2, 0, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 1, 2, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 2, 0, 2, 1, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 2, 0, 0, 3, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 2, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 2, 2, 0, 0, 2, 0, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58288>, <20, 1018880>, <24, 8026848>, <28, 39277568>, 
            <32, 143497868>, <36, 302698496>, <40, 548114784>, <44, 746369024>, <48, 786069904>, <52, 716277760>, <56, 
            493344192>, <60, 288077824>, <64, 142558822>, <68, 49631232>, <72, 22632896>, <76, 3944448>, <80, 2861712>, 
            <84, 183296>, <88, 294752>, <92, 5120>, <96, 22156>, <104, 1248>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 0, 2, 2, 2, 2, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 2, 0, 0, 2, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, 2, 0, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 1, 0, 2, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 2, 2, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 1, 0, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 1, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 1, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 3, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 0, 2, 0, 0, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 0, 2, 2, 0, 2, 2, 2, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 0, 2, 2, 2, 0, 2, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56624>, <20, 1028352>, <24, 8021856>, <28, 39292672>, 
            <32, 143454988>, <36, 302525440>, <40, 548433632>, <44, 746689536>, <48, 785414288>, <52, 716080640>, <56, 
            493900480>, <60, 288061952>, <64, 142432102>, <68, 49671168>, <72, 22549696>, <76, 3953664>, <80, 2902544>, 
            <84, 176384>, <88, 294368>, <92, 3840>, <96, 21772>, <104, 1120>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 0, 2, 2, 2, 0, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 2, 2, 0, 2, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 3, 2, 2, 0, 2, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 1, 2, 2, 2, 2, 0, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 3, 0, 0, 2, 2, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 2, 2, 1, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 2, 0, 3, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 0, 0, 0, 2, 1, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 2, 2, 2, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57168>, <20, 1028352>, <24, 8012352>, <28, 39292672>, 
            <32, 143503724>, <36, 302525440>, <40, 548312448>, <44, 746689536>, <48, 785582032>, <52, 716080640>, <56, 
            493770112>, <60, 288061952>, <64, 142480934>, <68, 49671168>, <72, 22548480>, <76, 3953664>, <80, 2898608>, 
            <84, 176384>, <88, 294464>, <92, 3840>, <96, 21996>, <104, 1152>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 0, 0, 2, 2, 0, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 0, 0, 2, 2, 0, 0, 2, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 1, 2, 2, 0, 0, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 3, 2, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 0, 1, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 2, 2, 0, 1, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 2, 2, 0, 1, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 2, 2, 2, 0, 0, 2, 2, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57888>, <20, 1019648>, <24, 8043024>, <28, 39276800>, 
            <32, 143404188>, <36, 302683136>, <40, 548352304>, <44, 746433536>, <48, 785755632>, <52, 716159488>, <56, 
            493550496>, <60, 288196096>, <64, 142531718>, <68, 49566720>, <72, 22585696>, <76, 3959808>, <80, 2887872>, 
            <84, 184064>, <88, 291664>, <92, 4352>, <96, 21852>, <104, 1136>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 0, 2, 0, 2, 2, 0, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 0, 2, 2, 2, 0, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 2, 0, 0, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 1, 2, 2, 0, 0, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 2, 3, 2, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 3, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 0, 1, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 1, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 0, 0, 2, 0, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56576>, <20, 1038080>, <24, 7947824>, <28, 39307520>, 
            <32, 143815996>, <36, 302347264>, <40, 547717008>, <44, 747031552>, <48, 786157744>, <52, 715844096>, <56, 
            493481696>, <60, 288085504>, <64, 142523334>, <68, 49689600>, <72, 22593568>, <76, 3968000>, <80, 2867488>, 
            <84, 169728>, <88, 296432>, <92, 2304>, <96, 24444>, <104, 1360>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 2, 2, 2, 0, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 1, 0, 2, 0, 0, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 1, 2, 0, 2, 0, 0, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 3, 2, 0, 0, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 3, 2, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 0, 1, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 2, 0, 3, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 2, 2, 0, 1, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 2, 0, 0, 0, 0, 2, 2, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 0, 0, 2, 0, 2, 2, 2, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 2, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58544>, <20, 1010944>, <24, 8074336>, <28, 39260928>, 
            <32, 143301772>, <36, 302840832>, <40, 548399840>, <44, 746177536>, <48, 785915792>, <52, 716238336>, <56, 
            493347008>, <60, 288330240>, <64, 142569062>, <68, 49462272>, <72, 22630592>, <76, 3965952>, <80, 2874256>, 
            <84, 191744>, <88, 289504>, <92, 4864>, <96, 21644>, <104, 1120>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 0, 0, 2, 2, 0, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 2, 2, 0, 0, 2, 2, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 2, 2, 0, 2, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 2, 1, 2, 2, 2, 2, 2, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 3, 0, 2, 0, 2, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 1, 2, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 0, 1, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 0, 0, 3, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 3, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 2, 2, 0, 2, 0, 0, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58944>, <20, 1012480>, <24, 8066224>, <28, 39259392>, 
            <32, 143344508>, <36, 302810112>, <40, 548293776>, <44, 746306560>, <48, 786059056>, <52, 716001792>, <56, 
            493242848>, <60, 288566784>, <64, 142599750>, <68, 49333248>, <72, 22636576>, <76, 3996672>, <80, 2869600>, 
            <84, 193280>, <88, 288880>, <92, 3328>, <96, 22204>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 0, 2, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 0, 2, 2, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 2, 0, 0, 0, 2, 2, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 3, 0, 0, 2, 0, 0, 0, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 2, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 1, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 2, 2, 3, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 1, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 2, 2, 2, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 2, 2, 2, 2, 0, 0, 2, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58336>, <20, 1019904>, <24, 8019216>, <28, 39276544>, 
            <32, 143545948>, <36, 302678016>, <40, 547986864>, <44, 746455040>, <48, 786250096>, <52, 716120064>, <56, 
            493206432>, <60, 288235520>, <64, 142607878>, <68, 49545216>, <72, 22630496>, <76, 3964928>, <80, 2861952>, 
            <84, 184320>, <88, 291920>, <92, 4096>, <96, 23068>, <104, 1264>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 2, 0, 0, 2, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 2, 2, 0, 0, 0, 2, 2, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 1, 2, 0, 0, 2, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 1, 2, 0, 0, 0, 2, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 3, 2, 2, 2, 2, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 2, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 2, 3, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 0, 0, 0, 2, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 0, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58080>, <20, 1019648>, <24, 8031952>, <28, 39276800>, 
            <32, 143470556>, <36, 302683136>, <40, 548180592>, <44, 746433536>, <48, 785988976>, <52, 716159488>, <56, 
            493386528>, <60, 288196096>, <64, 142570246>, <68, 49566720>, <72, 22604768>, <76, 3959808>, <80, 2876544>, 
            <84, 184064>, <88, 291600>, <92, 4352>, <96, 22428>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 2, 2, 2, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 2, 0, 0, 0, 2, 2, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 0, 0, 0, 0, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 2, 2, 0, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 3, 2, 0, 0, 0, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 2, 3, 2, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 0, 1, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 0, 2, 3, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 0, 2, 0, 3, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 2, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 0, 2, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 0, 0, 0, 2, 2, 2, 0, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57920>, <20, 1020160>, <24, 8035568>, <28, 39276288>, 
            <32, 143451388>, <36, 302672896>, <40, 548227024>, <44, 746476544>, <48, 785930544>, <52, 716080640>, <56, 
            493420128>, <60, 288274944>, <64, 142573382>, <68, 49523712>, <72, 22588576>, <76, 3970048>, <80, 2885472>, 
            <84, 184576>, <88, 289712>, <92, 3840>, <96, 22588>, <104, 1168>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 2, 2, 2, 0, 0, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 2, 0, 0, 2, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 2, 2, 0, 0, 2, 2, 2, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 2, 0, 2, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 1, 0, 0, 2, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 1, 0, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0, 2, 1, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 2, 3, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 2, 0, 0, 1, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 0, 2, 2, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58544>, <20, 1010432>, <24, 8067232>, <28, 39261440>, 
            <32, 143347212>, <36, 302851072>, <40, 548279840>, <44, 746134528>, <48, 786080144>, <52, 716317184>, <56, 
            493231424>, <60, 288251392>, <64, 142595942>, <68, 49505280>, <72, 22644032>, <76, 3955712>, <80, 2866576>, 
            <84, 191232>, <88, 289312>, <92, 5376>, <96, 22028>, <104, 1184>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 2, 2, 0, 0, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 2, 2, 0, 2, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 2, 2, 2, 0, 0, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 1, 0, 2, 2, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0, 3, 0, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 2, 2, 3, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 0, 0, 1, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 2, 2, 3, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 2, 2, 2, 0, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 2, 2, 2, 0, 2, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58272>, <20, 1019904>, <24, 8029008>, <28, 39276544>, 
            <32, 143485340>, <36, 302678016>, <40, 548143216>, <44, 746455040>, <48, 786043632>, <52, 716120064>, <56, 
            493338144>, <60, 288235520>, <64, 142596230>, <68, 49545216>, <72, 22596064>, <76, 3964928>, <80, 2879040>, 
            <84, 184320>, <88, 290704>, <92, 4096>, <96, 22620>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 2, 2, 2, 2, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 2, 0, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 0, 0, 2, 2, 2, 0, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 2, 2, 2, 0, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 1, 0, 0, 2, 2, 0, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 2, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 3, 0, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 2, 2, 3, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 2, 2, 2, 3, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 0, 0, 2, 0, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 2, 2, 2, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 2, 2, 2, 0, 0, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 0, 0, 2, 2, 2, 2, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57808>, <20, 1037824>, <24, 7912128>, <28, 39307776>, 
            <32, 144017132>, <36, 302352384>, <40, 547202304>, <44, 747010048>, <48, 786868432>, <52, 715883520>, <56, 
            492950144>, <60, 288046080>, <64, 142694694>, <68, 49711104>, <72, 22611712>, <76, 3962880>, <80, 2846512>, 
            <84, 169472>, <88, 295104>, <92, 2560>, <96, 25964>, <104, 1536>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 2, 2, 2, 2, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 2, 2, 0, 0, 2, 2, 0, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 3, 0, 2, 2, 2, 0, 0, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 2, 0, 0, 2, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 1, 2, 2, 0, 2, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 3, 2, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 0, 1, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 0, 2, 2, 0, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 2, 2, 2, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 2, 0, 2, 2, 0, 0, 0, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58160>, <20, 1019904>, <24, 8023008>, <28, 39276544>, 
            <32, 143525900>, <36, 302678016>, <40, 548035936>, <44, 746455040>, <48, 786186384>, <52, 716120064>, <56, 
            493247424>, <60, 288235520>, <64, 142603622>, <68, 49545216>, <72, 22619584>, <76, 3964928>, <80, 2868240>, 
            <84, 184320>, <88, 290912>, <92, 4096>, <96, 23052>, <104, 1248>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 2, 2, 2, 2, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 3, 0, 2, 0, 2, 0, 2, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 3, 0, 2, 2, 2, 2, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 1, 2, 2, 0, 0, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 2, 3, 2, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 0, 2, 1, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 0, 2, 0, 3, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 2, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 2, 2, 0, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 55936>, <20, 1037568>, <24, 7983728>, <28, 39308032>, 
            <32, 143601724>, <36, 302357504>, <40, 548268496>, <44, 746988544>, <48, 785415600>, <52, 715922944>, <56, 
            493989728>, <60, 288006656>, <64, 142422982>, <68, 49732608>, <72, 22515360>, <76, 3957760>, <80, 2910880>, 
            <84, 169216>, <88, 295216>, <92, 2816>, <96, 22652>, <104, 1168>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 0, 2, 0, 2, 2, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 3, 2, 0, 0, 0, 2, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 1, 0, 0, 2, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 3, 2, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 0, 1, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 2, 2, 2, 1, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 0, 0, 2, 0, 2, 2, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 0, 2, 0, 0, 2, 2, 0, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56176>, <20, 1037312>, <24, 7971168>, <28, 39308288>, 
            <32, 143676236>, <36, 302362624>, <40, 548077408>, <44, 746967040>, <48, 785671440>, <52, 715962368>, <56, 
            493817024>, <60, 287967232>, <64, 142453222>, <68, 49754112>, <72, 22546368>, <76, 3952640>, <80, 2893648>, 
            <84, 168960>, <88, 296416>, <92, 3072>, <96, 23116>, <104, 1248>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 2, 2, 2, 0, 0, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 0, 2, 2, 2, 0, 2, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 2, 0, 0, 0, 2, 0, 2, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 0, 2, 0, 2, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 0, 2, 2, 2, 0, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 1, 0, 0, 2, 2, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 3, 0, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 2, 2, 1, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 0, 2, 1, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 3, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 0, 0, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 2, 2, 2, 2, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57840>, <20, 1019392>, <24, 8037408>, <28, 39277056>, 
            <32, 143441484>, <36, 302688256>, <40, 548251680>, <44, 746412032>, <48, 785897488>, <52, 716198912>, <56, 
            493443648>, <60, 288156672>, <64, 142566886>, <68, 49588224>, <72, 22587200>, <76, 3954688>, <80, 2886096>, 
            <84, 183808>, <88, 290208>, <92, 4608>, <96, 22348>, <104, 1184>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 0, 2, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 2, 0, 0, 2, 2, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 2, 2, 0, 0, 0, 2, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 0, 0, 0, 2, 2, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 1, 2, 2, 0, 0, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 1, 0, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 3, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 2, 0, 0, 3, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 2, 2, 0, 2, 3, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 2, 0, 2, 0, 2, 2, 2, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58832>, <20, 1010944>, <24, 8076544>, <28, 39260928>, 
            <32, 143281260>, <36, 302840832>, <40, 548458048>, <44, 746177536>, <48, 785836752>, <52, 716238336>, <56, 
            493396736>, <60, 288330240>, <64, 142565926>, <68, 49462272>, <72, 22617984>, <76, 3965952>, <80, 2878768>, 
            <84, 191744>, <88, 290304>, <92, 4864>, <96, 21228>, <104, 1088>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 0, 0, 2, 0, 2, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 0, 0, 2, 2, 0, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 0, 2, 0, 0, 0, 2, 2, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 2, 2, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 3, 0, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 3, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 2, 1, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 2, 1, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 2, 0, 2, 0, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 0, 2, 2, 0, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58208>, <20, 1019904>, <24, 8029648>, <28, 39276544>, 
            <32, 143482460>, <36, 302678016>, <40, 548150896>, <44, 746455040>, <48, 786030192>, <52, 716120064>, <56, 
            493354272>, <60, 288235520>, <64, 142582790>, <68, 49545216>, <72, 22603744>, <76, 3964928>, <80, 2876160>, 
            <84, 184320>, <88, 291344>, <92, 4096>, <96, 22556>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 0, 0, 2, 2, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 2, 2, 2, 2, 0, 2, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 3, 2, 0, 0, 0, 2, 2, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 2, 3, 2, 0, 2, 2, 2, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 3, 0, 0, 0, 2, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 0, 3, 2, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 0, 3, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 2, 0, 2, 3, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 0, 2, 1, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 2, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58784>, <20, 1011200>, <24, 8068880>, <28, 39260672>, 
            <32, 143330844>, <36, 302835712>, <40, 548328752>, <44, 746199040>, <48, 786007280>, <52, 716198912>, <56, 
            493289888>, <60, 288369664>, <64, 142572422>, <68, 49440768>, <72, 22648160>, <76, 3971072>, <80, 2864704>, 
            <84, 192000>, <88, 290896>, <92, 4608>, <96, 21724>, <104, 1136>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 0, 0, 0, 2, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 2, 0, 2, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 0, 2, 2, 0, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 1, 0, 0, 2, 2, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 1, 2, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 2, 0, 3, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 2, 0, 1, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 2, 2, 3, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 0, 0, 2, 2, 2, 2, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 2, 2, 2, 0, 2, 0, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56480>, <20, 1038336>, <24, 7951824>, <28, 39307264>, 
            <32, 143793564>, <36, 302342144>, <40, 547770096>, <44, 747053056>, <48, 786098416>, <52, 715804672>, <56, 
            493499168>, <60, 288124928>, <64, 142554246>, <68, 49668096>, <72, 22555360>, <76, 3973120>, <80, 2885440>, 
            <84, 169984>, <88, 292880>, <92, 2048>, <96, 24668>, <104, 1328>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 2, 0, 2, 2, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 0, 2, 0, 0, 2, 0, 2, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 3, 2, 0, 2, 2, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 2, 0, 2, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 1, 2, 2, 2, 0, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 2, 3, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 0, 1, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 2, 0, 0, 2, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 2, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 2, 2, 2, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 2, 2, 0, 0, 2, 0, 0, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57248>, <20, 1029120>, <24, 8001360>, <28, 39291904>, 
            <32, 143571356>, <36, 302510080>, <40, 548139120>, <44, 746754048>, <48, 785808112>, <52, 715962368>, <56, 
            493631008>, <60, 288180224>, <64, 142485638>, <68, 49606656>, <72, 22591968>, <76, 3969024>, <80, 2878016>, 
            <84, 177152>, <88, 295824>, <92, 3072>, <96, 22620>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 0, 2, 2, 0, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 0, 2, 0, 2, 0, 2, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 3, 2, 0, 0, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 3, 2, 0, 0, 2, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 2, 3, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 2, 1, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0, 2, 2, 0, 1, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 0, 2, 2, 0, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 2, 2, 2, 2, 2, 0, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57504>, <20, 1020160>, <24, 8053968>, <28, 39276288>, 
            <32, 143341468>, <36, 302672896>, <40, 548518384>, <44, 746476544>, <48, 785510640>, <52, 716080640>, <56, 
            493762848>, <60, 288274944>, <64, 142424198>, <68, 49523712>, <72, 22618336>, <76, 3970048>, <80, 2878272>, 
            <84, 184576>, <88, 295696>, <92, 3840>, <96, 21084>, <104, 1072>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 2, 2, 0, 2, 2, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 0, 2, 2, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 2, 2, 0, 2, 0, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 0, 0, 2, 2, 0, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 1, 2, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 1, 0, 2, 0, 0, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 2, 2, 1, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 2, 2, 1, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 2, 2, 2, 0, 0, 2, 2, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 0, 0, 0, 2, 2, 0, 2, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58816>, <20, 1010944>, <24, 8074672>, <28, 39260928>, 
            <32, 143292668>, <36, 302840832>, <40, 548432016>, <44, 746177536>, <48, 785860144>, <52, 716238336>, <56, 
            493404128>, <60, 288330240>, <64, 142529862>, <68, 49462272>, <72, 22651936>, <76, 3965952>, <80, 2863840>, 
            <84, 191744>, <88, 293232>, <92, 4864>, <96, 21052>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 1, 2, 2, 2, 2, 0, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 3, 0, 0, 2, 0, 2, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 2, 2, 2, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 3, 2, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 3, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 0, 0, 2, 3, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 2, 2, 2, 1, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 2, 2, 2, 2, 2, 2, 0, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 0, 0, 0, 2, 0, 2, 2, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57664>, <20, 1020160>, <24, 8047280>, <28, 39276288>, 
            <32, 143382140>, <36, 302672896>, <40, 548406416>, <44, 746476544>, <48, 785683760>, <52, 716080640>, <56, 
            493600224>, <60, 288274944>, <64, 142521414>, <68, 49523712>, <72, 22577184>, <76, 3970048>, <80, 2893920>, 
            <84, 184576>, <88, 290416>, <92, 3840>, <96, 21948>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 0, 2, 0, 2, 0, 0, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 0, 0, 2, 0, 2, 2, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 3, 2, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 0, 1, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 2, 3, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 3, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 2, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 2, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56928>, <20, 1037312>, <24, 7934160>, <28, 39308288>, 
            <32, 143895388>, <36, 302362624>, <40, 547513200>, <44, 746967040>, <48, 786435184>, <52, 715962368>, <56, 
            493284128>, <60, 287967232>, <64, 142573062>, <68, 49754112>, <72, 22614496>, <76, 3952640>, <80, 2853376>, 
            <84, 168960>, <88, 297232>, <92, 3072>, <96, 24860>, <104, 1456>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 2, 0, 0, 2, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 0, 2, 2, 2, 2, 0, 2, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, 2, 0, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 3, 0, 0, 0, 0, 0, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 1, 2, 2, 0, 2, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 3, 2, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 2, 2, 3, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 2, 2, 0, 0, 0, 0, 0, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57552>, <20, 1020416>, <24, 8049408>, <28, 39276032>, 
            <32, 143371116>, <36, 302667776>, <40, 548433472>, <44, 746498048>, <48, 785647824>, <52, 716041216>, <56, 
            493625088>, <60, 288314368>, <64, 142516262>, <68, 49502208>, <72, 22572928>, <76, 3975168>, <80, 2896944>, 
            <84, 184832>, <88, 289792>, <92, 3584>, <96, 21996>, <104, 1088>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 2, 0, 2, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 2, 0, 2, 0, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 0, 2, 0, 2, 2, 0, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 2, 2, 2, 2, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 3, 2, 0, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 2, 0, 2, 0, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 3, 0, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 2, 1, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 2, 2, 0, 0, 1, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 2, 2, 0, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 2, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57328>, <20, 1028864>, <24, 7985312>, <28, 39292160>, 
            <32, 143672140>, <36, 302515200>, <40, 547874464>, <44, 746732544>, <48, 786169872>, <52, 716001792>, <56, 
            493376320>, <60, 288140800>, <64, 142545894>, <68, 49628160>, <72, 22620224>, <76, 3963904>, <80, 2862032>, 
            <84, 176896>, <88, 294944>, <92, 3328>, <96, 23628>, <104, 1312>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 0, 0, 2, 2, 0, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 0, 2, 0, 0, 0, 0, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 0, 0, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 2, 0, 2, 2, 2, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 1, 2, 0, 2, 2, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 3, 2, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 0, 1, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 2, 0, 1, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 0, 2, 2, 3, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 0, 2, 2, 0, 2, 0, 2, 0, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 2, 0, 0, 0, 0, 0, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58160>, <20, 1020672>, <24, 8029088>, <28, 39275776>, 
            <32, 143486604>, <36, 302662656>, <40, 548141600>, <44, 746519552>, <48, 786036368>, <52, 716001792>, <56, 
            493363008>, <60, 288353792>, <64, 142562406>, <68, 49480704>, <72, 22620480>, <76, 3980288>, <80, 2869776>, 
            <84, 185088>, <88, 292128>, <92, 3328>, <96, 22668>, <104, 1184>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 2, 0, 2, 2, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 2, 0, 2, 0, 2, 0, 0, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 0, 2, 2, 0, 2, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 2, 0, 2, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 1, 2, 0, 2, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 1, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 2, 3, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 2, 2, 2, 3, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 2, 2, 0, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 2, 2, 2, 0, 0, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57792>, <20, 1020416>, <24, 8035824>, <28, 39276032>, 
            <32, 143451772>, <36, 302667776>, <40, 548228048>, <44, 746498048>, <48, 785918000>, <52, 716041216>, <56, 
            493452384>, <60, 288314368>, <64, 142532166>, <68, 49502208>, <72, 22618272>, <76, 3975168>, <80, 2873568>, 
            <84, 184832>, <88, 292016>, <92, 3584>, <96, 22460>, <104, 1168>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 2, 2, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 0, 0, 0, 2, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 0, 2, 2, 2, 0, 0, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 3, 2, 0, 0, 2, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 2, 0, 2, 2, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 3, 0, 0, 2, 0, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 3, 0, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 1, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 0, 0, 3, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 2, 2, 2, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57440>, <20, 1028864>, <24, 8000464>, <28, 39292160>, 
            <32, 143573852>, <36, 302515200>, <40, 548130416>, <44, 746732544>, <48, 785834096>, <52, 716001792>, <56, 
            493582624>, <60, 288140800>, <64, 142540294>, <68, 49628160>, <72, 22554592>, <76, 3963904>, <80, 2892800>, 
            <84, 176896>, <88, 292880>, <92, 3328>, <96, 22812>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 0, 0, 0, 2, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 0, 2, 2, 2, 0, 0, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 3, 2, 2, 2, 2, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 2, 2, 2, 0, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 1, 2, 2, 0, 0, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 3, 0, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 2, 3, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 2, 2, 2, 3, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 0, 0, 2, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 0, 2, 0, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57232>, <20, 1029632>, <24, 7987264>, <28, 39291392>, 
            <32, 143661996>, <36, 302499840>, <40, 547898880>, <44, 746797056>, <48, 786139216>, <52, 715883520>, <56, 
            493393792>, <60, 288259072>, <64, 142548134>, <68, 49563648>, <72, 22610688>, <76, 3979264>, <80, 2867696>, 
            <84, 177664>, <88, 293440>, <92, 2560>, <96, 23852>, <104, 1280>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 2, 0, 0, 2, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 0, 2, 0, 2, 2, 2, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 0, 2, 2, 0, 2, 2, 0, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 1, 0, 2, 0, 0, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 1, 2, 0, 0, 2, 0, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 1, 0, 2, 0, 2, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 1, 2, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 3, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 2, 2, 0, 1, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 1, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 0, 2, 2, 0, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 2, 2, 2, 0, 0, 2, 0, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56720>, <20, 1028864>, <24, 8023936>, <28, 39292160>, 
            <32, 143438124>, <36, 302515200>, <40, 548486208>, <44, 746732544>, <48, 785323600>, <52, 716001792>, <56, 
            493998592>, <60, 288140800>, <64, 142359974>, <68, 49628160>, <72, 22588800>, <76, 3963904>, <80, 2886128>, 
            <84, 176896>, <88, 299136>, <92, 3328>, <96, 21164>, <104, 1088>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 2, 2, 2, 2, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 0, 2, 2, 0, 0, 0, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 3, 0, 2, 2, 2, 2, 0, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 1, 0, 0, 2, 2, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 2, 3, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 0, 2, 0, 1, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 0, 0, 2, 1, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 2, 0, 2, 2, 2, 0, 0, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 2, 0, 0, 2, 0, 2, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 55936>, <20, 1038336>, <24, 7989808>, <28, 39307264>, 
            <32, 143562428>, <36, 302342144>, <40, 548374160>, <44, 747053056>, <48, 785265584>, <52, 715804672>, <56, 
            494105312>, <60, 288124928>, <64, 142381766>, <68, 49668096>, <72, 22516256>, <76, 3973120>, <80, 2912416>, 
            <84, 169984>, <88, 296432>, <92, 2048>, <96, 22268>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 0, 2, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 0, 2, 2, 0, 0, 2, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 2, 0, 0, 2, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 1, 0, 0, 2, 2, 0, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 2, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 0, 3, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 0, 2, 3, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 3, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 2, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 0, 0, 0, 0, 2, 2, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58448>, <20, 1028608>, <24, 7952768>, <28, 39292416>, 
            <32, 143856108>, <36, 302520320>, <40, 547401152>, <44, 746711040>, <48, 786830288>, <52, 716041216>, <56, 
            492869632>, <60, 288101376>, <64, 142726438>, <68, 49649664>, <72, 22619776>, <76, 3958784>, <80, 2850224>, 
            <84, 176640>, <88, 291968>, <92, 3584>, <96, 25196>, <104, 1472>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 2, 2, 2, 2, 0, 0, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 3, 2, 2, 0, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 0, 2, 2, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 1, 0, 0, 2, 2, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 0, 1, 2, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 2, 0, 3, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 0, 0, 2, 3, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 2, 2, 3, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 2, 0, 2, 0, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 2, 2, 0, 0, 2, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 2, 0, 2, 2, 0, 2, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57776>, <20, 1020416>, <24, 8043104>, <28, 39276032>, 
            <32, 143405452>, <36, 302667776>, <40, 548350688>, <44, 746498048>, <48, 785748368>, <52, 716041216>, <56, 
            493575360>, <60, 288314368>, <64, 142497894>, <68, 49502208>, <72, 22610112>, <76, 3975168>, <80, 2878608>, 
            <84, 184832>, <88, 293088>, <92, 3584>, <96, 21900>, <104, 1120>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 2, 2, 0, 2, 2, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 0, 2, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 2, 0, 0, 2, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 3, 2, 0, 0, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 3, 0, 0, 2, 0, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 1, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 2, 1, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 3, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 2, 0, 0, 2, 2, 0, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57888>, <20, 1019904>, <24, 8032848>, <28, 39276544>, 
            <32, 143468060>, <36, 302678016>, <40, 548189296>, <44, 746455040>, <48, 785962992>, <52, 716120064>, <56, 
            493434912>, <60, 288235520>, <64, 142515590>, <68, 49545216>, <72, 22642144>, <76, 3964928>, <80, 2861760>, 
            <84, 184320>, <88, 294544>, <92, 4096>, <96, 22236>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 0, 0, 2, 2, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 2, 2, 2, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 2, 0, 0, 0, 0, 2, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 2, 2, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 3, 0, 0, 0, 2, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 3, 2, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 2, 0, 1, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 2, 3, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 2, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 2, 2, 0, 2, 0, 2, 0, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 2, 0, 2, 2, 2, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58016>, <20, 1019648>, <24, 8032592>, <28, 39276800>, 
            <32, 143467676>, <36, 302683136>, <40, 548188272>, <44, 746433536>, <48, 785975536>, <52, 716159488>, <56, 
            493402656>, <60, 288196096>, <64, 142556806>, <68, 49566720>, <72, 22612448>, <76, 3959808>, <80, 2873664>, 
            <84, 184064>, <88, 292240>, <92, 4352>, <96, 22364>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 0, 0, 2, 2, 0, 2, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 2, 0, 2, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 1, 0, 2, 2, 0, 2, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 1, 2, 2, 2, 2, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 2, 1, 2, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 0, 3, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 1, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 2, 0, 2, 2, 0, 2, 2, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 2, 2, 0, 2, 2, 0, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57696>, <20, 1028096>, <24, 7973520>, <28, 39292928>, 
            <32, 143740124>, <36, 302530560>, <40, 547696688>, <44, 746668032>, <48, 786423920>, <52, 716120064>, <56, 
            493171360>, <60, 288022528>, <64, 142631686>, <68, 49692672>, <72, 22607200>, <76, 3948544>, <80, 2862848>, 
            <84, 176128>, <88, 292816>, <92, 4096>, <96, 24220>, <104, 1392>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 2, 0, 0, 0, 2, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 0, 2, 2, 0, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 0, 2, 0, 2, 0, 0, 2, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 0, 0, 2, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 1, 2, 0, 0, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 3, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 2, 2, 3, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 2, 0, 1, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 1, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 2, 0, 2, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 2, 2, 2, 0, 2, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 2, 2, 0, 0, 2, 2, 0, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58512>, <20, 1020928>, <24, 8020480>, <28, 39275520>, 
            <32, 143532844>, <36, 302657536>, <40, 548029120>, <44, 746541056>, <48, 786178128>, <52, 715962368>, <56, 
            493281024>, <60, 288393216>, <64, 142556582>, <68, 49459200>, <72, 22656640>, <76, 3985408>, <80, 2851056>, 
            <84, 185344>, <88, 295168>, <92, 3072>, <96, 22700>, <104, 1216>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 2, 2, 2, 0, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 1, 2, 0, 0, 0, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 3, 0, 2, 2, 2, 2, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 3, 2, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 0, 1, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 0, 2, 3, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2, 3, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 0, 2, 2, 0, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 0, 2, 2, 2, 0, 0, 2, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 59008>, <20, 1009920>, <24, 8058544>, <28, 39261952>, 
            <32, 143392188>, <36, 302861312>, <40, 548168976>, <44, 746091520>, <48, 786229168>, <52, 716396032>, <56, 
            493124576>, <60, 288172544>, <64, 142623942>, <68, 49548288>, <72, 22655776>, <76, 3945472>, <80, 2857120>, 
            <84, 190720>, <88, 290928>, <92, 5888>, <96, 22012>, <104, 1232>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 0, 2, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 2, 2, 2, 0, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 3, 2, 0, 0, 2, 0, 0, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 2, 0, 2, 0, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 1, 2, 2, 2, 0, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 1, 2, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 0, 3, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 0, 1, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 2, 2, 2, 3, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 0, 2, 0, 0, 2, 2, 2, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56336>, <20, 1046272>, <24, 7910336>, <28, 39323904>, 
            <32, 143949100>, <36, 302199808>, <40, 547592320>, <44, 747244544>, <48, 786109776>, <52, 715844096>, <56, 
            493587072>, <60, 287872512>, <64, 142536614>, <68, 49837056>, <72, 22534144>, <76, 3951616>, <80, 2883696>, 
            <84, 161536>, <88, 297920>, <92, 2304>, <96, 24748>, <104, 1408>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 2, 2, 0, 2, 2, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 0, 0, 2, 2, 2, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 0, 2, 2, 2, 0, 0, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 2, 2, 2, 2, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 3, 2, 2, 0, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 1, 2, 2, 2, 0, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 2, 1, 0, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 1, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 2, 0, 2, 3, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 0, 2, 2, 0, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 2, 0, 0, 2, 2, 0, 2, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 55568>, <20, 1045760>, <24, 7938368>, <28, 39324416>, 
            <32, 143786796>, <36, 302210048>, <40, 548010496>, <44, 747201536>, <48, 785533776>, <52, 715922944>, <56, 
            494011776>, <60, 287793664>, <64, 142407590>, <68, 49880064>, <72, 22513408>, <76, 3941376>, <80, 2901360>, 
            <84, 161024>, <88, 299840>, <92, 2816>, <96, 23212>, <104, 1280>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 0, 0, 2, 2, 2, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 0, 0, 2, 0, 0, 2, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 2, 2, 2, 2, 2, 0, 2, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 3, 2, 2, 2, 2, 2, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 2, 0, 0, 2, 2, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 1, 0, 0, 0, 2, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 3, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 2, 2, 2, 3, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 0, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56704>, <20, 1037312>, <24, 7940464>, <28, 39308288>, 
            <32, 143861052>, <36, 302362624>, <40, 547595984>, <44, 746967040>, <48, 786334640>, <52, 715962368>, <56, 
            493333856>, <60, 287967232>, <64, 142591430>, <68, 49754112>, <72, 22577312>, <76, 3952640>, <80, 2871712>, 
            <84, 168960>, <88, 293936>, <92, 3072>, <96, 24956>, <104, 1424>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 0, 2, 2, 0, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 2, 2, 2, 0, 0, 2, 0, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 3, 0, 0, 2, 0, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 3, 0, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 3, 0, 2, 0, 0, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 3, 2, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 0, 3, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 2, 3, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 3, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 2, 0, 2, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 59104>, <20, 1011456>, <24, 8055504>, <28, 39260416>, 
            <32, 143409116>, <36, 302830592>, <40, 548127344>, <44, 746220544>, <48, 786281840>, <52, 716159488>, <56, 
            493093664>, <60, 288409088>, <64, 142623494>, <68, 49419264>, <72, 22666208>, <76, 3976192>, <80, 2852992>, 
            <84, 192256>, <88, 290576>, <92, 4352>, <96, 22428>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 0, 2, 0, 0, 2, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 2, 0, 2, 2, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 3, 0, 2, 0, 0, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 2, 2, 2, 2, 0, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 3, 2, 2, 2, 0, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 3, 0, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 1, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 2, 2, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 0, 2, 2, 2, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 0, 2, 2, 2, 0, 0, 0, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57824>, <20, 1028864>, <24, 7969168>, <28, 39292160>, 
            <32, 143764316>, <36, 302515200>, <40, 547638320>, <44, 746732544>, <48, 786493808>, <52, 716001792>, <56, 
            493139104>, <60, 288140800>, <64, 142615558>, <68, 49628160>, <72, 22634848>, <76, 3963904>, <80, 2850176>, 
            <84, 176896>, <88, 294608>, <92, 3328>, <96, 24348>, <104, 1392>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 0, 0, 0, 2, 2, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 0, 2, 2, 2, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 2, 0, 2, 0, 2, 0, 2, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 1, 0, 2, 0, 0, 2, 0, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 1, 0, 2, 2, 0, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 3, 0, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 1, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 2, 2, 1, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 2, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58752>, <20, 1011200>, <24, 8064112>, <28, 39260672>, 
            <32, 143362876>, <36, 302835712>, <40, 548239824>, <44, 746199040>, <48, 786140080>, <52, 716198912>, <56, 
            493175648>, <60, 288369664>, <64, 142629318>, <68, 49440768>, <72, 22630048>, <76, 3971072>, <80, 2871712>, 
            <84, 192000>, <88, 287536>, <92, 4608>, <96, 22396>, <104, 1168>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 2, 0, 2, 0, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 0, 0, 2, 2, 2, 0, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 3, 0, 2, 2, 0, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 3, 2, 2, 2, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 3, 0, 2, 2, 0, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 3, 0, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 0, 2, 1, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 0, 1, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 1, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 2, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58960>, <20, 1011456>, <24, 8082368>, <28, 39260416>, 
            <32, 143241580>, <36, 302830592>, <40, 548562688>, <44, 746220544>, <48, 785699280>, <52, 716159488>, <56, 
            493480064>, <60, 288409088>, <64, 142565926>, <68, 49419264>, <72, 22589184>, <76, 3976192>, <80, 2892208>, 
            <84, 192256>, <88, 289216>, <92, 4352>, <96, 20972>, <104, 1024>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 2, 2, 2, 2, 0, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 2, 2, 0, 2, 2, 2, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 3, 0, 0, 2, 2, 2, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 3, 2, 2, 2, 0, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 3, 2, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 0, 0, 3, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 0, 1, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 2, 2, 2, 2, 1, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 2, 2, 0, 2, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 0, 0, 0, 0, 2, 0, 2, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 2, 0, 2, 2, 2, 2, 0, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57568>, <20, 1028608>, <24, 7981904>, <28, 39292416>, 
            <32, 143688924>, <36, 302520320>, <40, 547832048>, <44, 746711040>, <48, 786232688>, <52, 716041216>, <56, 
            493319200>, <60, 288101376>, <64, 142577926>, <68, 49649664>, <72, 22609120>, <76, 3958784>, <80, 2864768>, 
            <84, 176640>, <88, 294288>, <92, 3584>, <96, 23708>, <104, 1328>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 0, 0, 0, 2, 0, 2, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 2, 2, 2, 2, 0, 2, 2, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 3, 2, 2, 2, 0, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 2, 2, 2, 0, 0, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 3, 0, 0, 2, 2, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 1, 0, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 2, 3, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 3, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58496>, <20, 1011456>, <24, 8074800>, <28, 39260416>, 
            <32, 143299772>, <36, 302830592>, <40, 548404880>, <44, 746220544>, <48, 785907632>, <52, 716159488>, <56, 
            493355744>, <60, 288409088>, <64, 142563014>, <68, 49419264>, <72, 22632992>, <76, 3976192>, <80, 2874016>, 
            <84, 192256>, <88, 289264>, <92, 4352>, <96, 21756>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 0, 0, 2, 2, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 2, 2, 0, 2, 0, 2, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 2, 0, 2, 0, 0, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 2, 2, 2, 0, 2, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 1, 2, 2, 2, 2, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 2, 1, 2, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 3, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 0, 0, 0, 1, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 2, 2, 3, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 2, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 2, 2, 2, 2, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 2, 0, 0, 2, 2, 2, 0, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57680>, <20, 1028096>, <24, 7980800>, <28, 39292928>, 
            <32, 143693804>, <36, 302530560>, <40, 547819328>, <44, 746668032>, <48, 786254288>, <52, 716120064>, <56, 
            493294336>, <60, 288022528>, <64, 142597414>, <68, 49692672>, <72, 22599040>, <76, 3948544>, <80, 2867888>, 
            <84, 176128>, <88, 293888>, <92, 4096>, <96, 23660>, <104, 1344>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 2, 0, 0, 2, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 0, 0, 0, 2, 0, 2, 2, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 3, 2, 2, 2, 0, 0, 2, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 1, 2, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 2, 3, 0, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 2, 2, 1, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 2, 1, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 3, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 2, 0, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 2, 2, 0, 0, 2, 2, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56960>, <20, 1037568>, <24, 7956208>, <28, 39308032>, 
            <32, 143754044>, <36, 302357504>, <40, 547885136>, <44, 746988544>, <48, 785930672>, <52, 715922944>, <56, 
            493629536>, <60, 288006656>, <64, 142505414>, <68, 49732608>, <72, 22562720>, <76, 3957760>, <80, 2880160>, 
            <84, 169216>, <88, 297904>, <92, 2816>, <96, 23420>, <104, 1296>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 3, 0, 0, 2, 0, 2, 0, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 2, 2, 0, 2, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 1, 2, 2, 0, 2, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 1, 2, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 0, 0, 1, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 2, 2, 0, 1, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 1, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 0, 2, 2, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 2, 2, 2, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 2, 0, 2, 2, 0, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57568>, <20, 1020160>, <24, 8053328>, <28, 39276288>, 
            <32, 143344348>, <36, 302672896>, <40, 548510704>, <44, 746476544>, <48, 785524080>, <52, 716080640>, <56, 
            493746720>, <60, 288274944>, <64, 142437638>, <68, 49523712>, <72, 22610656>, <76, 3970048>, <80, 2881152>, 
            <84, 184576>, <88, 295056>, <92, 3840>, <96, 21148>, <104, 1072>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 0, 2, 2, 2, 0, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 0, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 2, 0, 0, 2, 2, 0, 2, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 3, 2, 2, 0, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 2, 2, 2, 2, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 2, 3, 2, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 0, 2, 2, 1, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 2, 2, 0, 0, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 2, 2, 0, 0, 2, 2, 0, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 2, 2, 0, 2, 0, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57424>, <20, 1029376>, <24, 8005696>, <28, 39291648>, 
            <32, 143539820>, <36, 302504960>, <40, 548224384>, <44, 746775552>, <48, 785693136>, <52, 715922944>, <56, 
            493705600>, <60, 288219648>, <64, 142477350>, <68, 49585152>, <72, 22575104>, <76, 3974144>, <80, 2885552>, 
            <84, 177408>, <88, 296000>, <92, 2816>, <96, 22252>, <104, 1152>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 0, 2, 2, 0, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 2, 2, 2, 0, 0, 0, 2, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 2, 2, 0, 2, 2, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 0, 2, 2, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 3, 0, 2, 2, 2, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 3, 2, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 0, 0, 1, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 2, 3, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 0, 0, 2, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 2, 2, 0, 2, 2, 0, 0, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58400>, <20, 1020160>, <24, 8017552>, <28, 39276288>, 
            <32, 143554972>, <36, 302672896>, <40, 547964848>, <44, 746476544>, <48, 786277872>, <52, 716080640>, <56, 
            493190304>, <60, 288274944>, <64, 142606982>, <68, 49523712>, <72, 22637152>, <76, 3970048>, <80, 2858688>, 
            <84, 184576>, <88, 292304>, <92, 3840>, <96, 23132>, <104, 1264>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 2, 2, 2, 0, 0, 2, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 3, 0, 0, 2, 2, 2, 0, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 3, 0, 2, 2, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 3, 0, 2, 2, 0, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 3, 2, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 0, 1, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 0, 0, 0, 3, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 2, 2, 0, 3, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57216>, <20, 1037568>, <24, 7944496>, <28, 39308032>, 
            <32, 143823292>, <36, 302357504>, <40, 547705744>, <44, 746988544>, <48, 786177456>, <52, 715922944>, <56, 
            493449440>, <60, 288006656>, <64, 142557382>, <68, 49732608>, <72, 22574112>, <76, 3957760>, <80, 2871712>, 
            <84, 169216>, <88, 297200>, <92, 2816>, <96, 24060>, <104, 1360>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 0, 2, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 1, 0, 2, 0, 2, 2, 0, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 3, 2, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 2, 1, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 3, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 0, 0, 2, 1, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 2, 2, 2, 0, 2, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57328>, <20, 1029120>, <24, 7993440>, <28, 39291904>, 
            <32, 143620556>, <36, 302510080>, <40, 548008800>, <44, 746754048>, <48, 785991184>, <52, 715962368>, <56, 
            493491904>, <60, 288180224>, <64, 142533350>, <68, 49606656>, <72, 22592448>, <76, 3969024>, <80, 2875856>, 
            <84, 177152>, <88, 294112>, <92, 3072>, <96, 23244>, <104, 1248>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 0, 0, 0, 2, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 3, 2, 2, 0, 0, 2, 0, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 0, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 1, 2, 0, 0, 0, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 2, 2, 1, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 0, 2, 1, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 2, 2, 0, 0, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 0, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57584>, <20, 1029120>, <24, 7981728>, <28, 39291904>, 
            <32, 143689804>, <36, 302510080>, <40, 547829408>, <44, 746754048>, <48, 786237968>, <52, 715962368>, <56, 
            493311808>, <60, 288180224>, <64, 142585318>, <68, 49606656>, <72, 22603840>, <76, 3969024>, <80, 2867408>, 
            <84, 177152>, <88, 293408>, <92, 3072>, <96, 23884>, <104, 1312>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 0, 0, 2, 0, 2, 2, 0, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 1, 2, 2, 2, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 2, 3, 0, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0, 2, 3, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 0, 2, 3, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 1, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 2, 2, 0, 0, 2, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 2, 0, 2, 2, 0, 0, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58160>, <20, 1019648>, <24, 8024032>, <28, 39276800>, 
            <32, 143519756>, <36, 302683136>, <40, 548050272>, <44, 746433536>, <48, 786172048>, <52, 716159488>, <56, 
            493247424>, <60, 288196096>, <64, 142617958>, <68, 49566720>, <72, 22605248>, <76, 3959808>, <80, 2874384>, 
            <84, 184064>, <88, 289888>, <92, 4352>, <96, 23052>, <104, 1248>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 0, 0, 0, 2, 2, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 0, 2, 2, 0, 0, 0, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 1, 2, 2, 0, 0, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 3, 0, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 1, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 0, 2, 1, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 2, 2, 2, 3, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 2, 2, 2, 2, 0, 2, 2, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58480>, <20, 1011200>, <24, 8073952>, <28, 39260672>, 
            <32, 143305036>, <36, 302835712>, <40, 548393184>, <44, 746199040>, <48, 785916688>, <52, 716198912>, <56, 
            493363136>, <60, 288369664>, <64, 142541286>, <68, 49440768>, <72, 22652608>, <76, 3971072>, <80, 2865232>, 
            <84, 192000>, <88, 291168>, <92, 4608>, <96, 21580>, <104, 1120>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 0, 2, 0, 2, 2, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 3, 0, 2, 2, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 0, 3, 0, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 2, 2, 3, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 2, 2, 0, 1, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 0, 2, 2, 3, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 2, 2, 2, 2, 2, 0, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 0, 0, 2, 2, 2, 2, 0, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 0, 2, 2, 2, 2, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58464>, <20, 1011200>, <24, 8071056>, <28, 39260672>, 
            <32, 143325660>, <36, 302835712>, <40, 548330288>, <44, 746199040>, <48, 786026096>, <52, 716198912>, <56, 
            493241504>, <60, 288369664>, <64, 142634246>, <68, 49440768>, <72, 22600544>, <76, 3971072>, <80, 2887168>, 
            <84, 192000>, <88, 284880>, <92, 4608>, <96, 22428>, <104, 1136>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 2, 2, 0, 0, 0, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 0, 0, 0, 2, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 0, 2, 0, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 3, 2, 2, 2, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 3, 2, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 0, 1, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 0, 0, 3, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 3, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 2, 2, 2, 2, 0, 2, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 2, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56800>, <20, 1037056>, <24, 7954768>, <28, 39308544>, 
            <32, 143768028>, <36, 302367744>, <40, 547840240>, <44, 746945536>, <48, 786007920>, <52, 716001792>, <56, 
            493547552>, <60, 287927808>, <64, 142564102>, <68, 49775616>, <72, 22531296>, <76, 3947520>, <80, 2893696>, 
            <84, 168704>, <88, 293776>, <92, 3328>, <96, 23964>, <104, 1328>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 0, 2, 0, 2, 0, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 0, 2, 0, 0, 0, 0, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 3, 2, 0, 2, 2, 2, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 3, 0, 2, 2, 0, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 2, 3, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 0, 1, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 2, 0, 2, 0, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 0, 2, 2, 0, 0, 2, 2, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57904>, <20, 1019904>, <24, 8034720>, <28, 39276544>, 
            <32, 143456652>, <36, 302678016>, <40, 548215328>, <44, 746455040>, <48, 785939600>, <52, 716120064>, <56, 
            493427520>, <60, 288235520>, <64, 142551654>, <68, 49545216>, <72, 22608192>, <76, 3964928>, <80, 2876688>, 
            <84, 184320>, <88, 291616>, <92, 4096>, <96, 22412>, <104, 1184>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 2, 2, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 2, 0, 0, 2, 2, 2, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 3, 0, 0, 2, 0, 0, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 3, 0, 0, 2, 0, 0, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 2, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 1, 2, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 0, 1, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 3, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 2, 2, 2, 1, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 2, 2, 2, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57088>, <20, 1037056>, <24, 7947824>, <28, 39308544>, 
            <32, 143805244>, <36, 302367744>, <40, 547749776>, <44, 746945536>, <48, 786121904>, <52, 716001792>, <56, 
            493481696>, <60, 287927808>, <64, 142559174>, <68, 49775616>, <72, 22560800>, <76, 3947520>, <80, 2878240>, 
            <84, 168704>, <88, 296432>, <92, 3328>, <96, 23932>, <104, 1360>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 2, 0, 0, 2, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 0, 2, 0, 2, 0, 0, 2, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 2, 0, 2, 2, 0, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 2, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 2, 0, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 2, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 2, 0, 1, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 0, 0, 0, 3, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 2, 2, 3, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 0, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 0, 2, 2, 2, 0, 0, 0, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57952>, <20, 1020160>, <24, 8040336>, <28, 39276288>, 
            <32, 143419356>, <36, 302672896>, <40, 548315952>, <44, 746476544>, <48, 785797744>, <52, 716080640>, <56, 
            493534368>, <60, 288274944>, <64, 142516486>, <68, 49523712>, <72, 22606688>, <76, 3970048>, <80, 2878464>, 
            <84, 184576>, <88, 293072>, <92, 3840>, <96, 21916>, <104, 1136>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 2, 2, 0, 2, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 2, 2, 2, 2, 0, 2, 0, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 0, 0, 2, 2, 0, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 0, 2, 0, 0, 0, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 3, 2, 2, 0, 0, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 2, 1, 2, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 3, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 0, 1, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 2, 0, 2, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 0, 2, 2, 0, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 0, 0, 0, 0, 2, 2, 0, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57744>, <20, 1018624>, <24, 8046528>, <28, 39277824>, 
            <32, 143385260>, <36, 302703616>, <40, 548398976>, <44, 746347520>, <48, 785694800>, <52, 716317184>, <56, 
            493590144>, <60, 288038400>, <64, 142526118>, <68, 49652736>, <72, 22577664>, <76, 3939328>, <80, 2891760>, 
            <84, 183040>, <88, 291776>, <92, 5376>, <96, 21548>, <104, 1152>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 2, 2, 0, 2, 2, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 2, 0, 0, 2, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 2, 2, 0, 2, 2, 0, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 3, 0, 0, 0, 0, 0, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 1, 2, 2, 2, 0, 2, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 1, 0, 0, 0, 2, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 1, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 1, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 2, 0, 0, 3, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 3, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 2, 0, 2, 0, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57888>, <20, 1021184>, <24, 8035856>, <28, 39275264>, 
            <32, 143450268>, <36, 302652416>, <40, 548229424>, <44, 746562560>, <48, 785927664>, <52, 715922944>, <56, 
            493421472>, <60, 288432640>, <64, 142574726>, <68, 49437696>, <72, 22585696>, <76, 3990528>, <80, 2887872>, 
            <84, 185600>, <88, 288592>, <92, 2816>, <96, 22876>, <104, 1136>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 0, 2, 2, 2, 0, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 0, 2, 0, 2, 0, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 0, 2, 0, 2, 2, 0, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 2, 0, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 3, 2, 2, 2, 2, 2, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 2, 1, 0, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 2, 2, 3, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 0, 1, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 2, 2, 0, 0, 2, 0, 0, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57936>, <20, 1019904>, <24, 8029312>, <28, 39276544>, 
            <32, 143491564>, <36, 302678016>, <40, 548118720>, <44, 746455040>, <48, 786085840>, <52, 716120064>, <56, 
            493297152>, <60, 288235520>, <64, 142621990>, <68, 49545216>, <72, 22582400>, <76, 3964928>, <80, 2886576>, 
            <84, 184320>, <88, 287616>, <92, 4096>, <96, 23148>, <104, 1216>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 2, 2, 0, 2, 2, 2, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 0, 0, 2, 0, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 1, 0, 2, 0, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 2, 2, 2, 0, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 3, 0, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 3, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 0, 2, 0, 3, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 0, 2, 0, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 0, 2, 2, 0, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56912>, <20, 1036800>, <24, 7943488>, <28, 39308800>, 
            <32, 143836780>, <36, 302372864>, <40, 547664512>, <44, 746924032>, <48, 786236880>, <52, 716041216>, <56, 
            493407104>, <60, 287888384>, <64, 142567462>, <68, 49797120>, <72, 22577664>, <76, 3942400>, <80, 2870704>, 
            <84, 168448>, <88, 296256>, <92, 3584>, <96, 24300>, <104, 1408>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 2, 0, 2, 0, 0, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 0, 2, 0, 2, 2, 2, 0, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 3, 2, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 2, 2, 2, 0, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 1, 0, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 2, 3, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 0, 0, 2, 3, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 0, 2, 0, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 2, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 2, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 0, 0, 0, 2, 2, 2, 0, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58272>, <20, 1010944>, <24, 8102480>, <28, 39260928>, 
            <32, 143128476>, <36, 302840832>, <40, 548850544>, <44, 746177536>, <48, 785306352>, <52, 716238336>, <56, 
            493765664>, <60, 288330240>, <64, 142484614>, <68, 49462272>, <72, 22568928>, <76, 3965952>, <80, 2907712>, 
            <84, 191744>, <88, 289424>, <92, 4864>, <96, 20060>, <104, 944>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 0, 2, 2, 0, 2, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 2, 0, 2, 2, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 0, 0, 0, 2, 2, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 2, 0, 0, 0, 0, 2, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 1, 2, 2, 0, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 3, 2, 0, 0, 2, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 1, 0, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 1, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 2, 3, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 3, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 0, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 0, 2, 2, 0, 2, 0, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57040>, <20, 1028864>, <24, 8020736>, <28, 39292160>, 
            <32, 143452524>, <36, 302515200>, <40, 548447808>, <44, 746732544>, <48, 785390800>, <52, 716001792>, <56, 
            493917952>, <60, 288140800>, <64, 142427174>, <68, 49628160>, <72, 22550400>, <76, 3963904>, <80, 2900528>, 
            <84, 176896>, <88, 295936>, <92, 3328>, <96, 21484>, <104, 1088>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 0, 2, 2, 2, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 0, 2, 0, 2, 0, 0, 0, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0, 2, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 3, 2, 2, 2, 0, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 3, 2, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 2, 2, 1, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 0, 2, 0, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 2, 2, 0, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56912>, <20, 1037312>, <24, 7950592>, <28, 39308288>, 
            <32, 143791340>, <36, 302362624>, <40, 547784512>, <44, 746967040>, <48, 786072528>, <52, 715962368>, <56, 
            493522688>, <60, 287967232>, <64, 142540582>, <68, 49754112>, <72, 22564224>, <76, 3952640>, <80, 2878384>, 
            <84, 168960>, <88, 296448>, <92, 3072>, <96, 23916>, <104, 1344>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 2, 0, 0, 2, 2, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 0, 2, 2, 0, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 2, 0, 2, 0, 2, 0, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 0, 2, 2, 0, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 2, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 1, 0, 0, 2, 2, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 3, 0, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 2, 2, 3, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 0, 2, 3, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 2, 2, 2, 3, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 2, 2, 2, 0, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 2, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57728>, <20, 1019904>, <24, 8038512>, <28, 39276544>, 
            <32, 143436604>, <36, 302678016>, <40, 548264400>, <44, 746455040>, <48, 785875888>, <52, 716120064>, <56, 
            493468512>, <60, 288235520>, <64, 142547398>, <68, 49545216>, <72, 22597280>, <76, 3964928>, <80, 2882976>, 
            <84, 184320>, <88, 290608>, <92, 4096>, <96, 22396>, <104, 1168>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 0, 2, 0, 0, 2, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 2, 2, 2, 2, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 1, 2, 2, 2, 2, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 2, 0, 2, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 3, 0, 0, 0, 2, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 0, 1, 0, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 2, 2, 3, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 2, 2, 3, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 2, 0, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 0, 2, 0, 0, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 2, 2, 0, 2, 2, 0, 0, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57632>, <20, 1020672>, <24, 8040464>, <28, 39275776>, 
            <32, 143426460>, <36, 302662656>, <40, 548288816>, <44, 746519552>, <48, 785845232>, <52, 716001792>, <56, 
            493485984>, <60, 288353792>, <64, 142549638>, <68, 49480704>, <72, 22587744>, <76, 3980288>, <80, 2888640>, 
            <84, 185088>, <88, 289104>, <92, 3328>, <96, 22620>, <104, 1136>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 0, 0, 2, 2, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 3, 2, 0, 2, 2, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 2, 2, 0, 2, 2, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 3, 0, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 1, 0, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 1, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 0, 0, 1, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 2, 2, 0, 1, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 2, 2, 2, 0, 2, 2, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 2, 2, 0, 0, 0, 2, 2, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56864>, <20, 1029120>, <24, 8023504>, <28, 39291904>, 
            <32, 143438620>, <36, 302510080>, <40, 548482544>, <44, 746754048>, <48, 785341424>, <52, 715962368>, <56, 
            493958944>, <60, 288180224>, <64, 142408582>, <68, 49606656>, <72, 22553824>, <76, 3969024>, <80, 2900672>, 
            <84, 177152>, <88, 295952>, <92, 3072>, <96, 21468>, <104, 1072>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 0, 2, 0, 2, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 2, 0, 2, 2, 2, 2, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 1, 0, 2, 2, 0, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 1, 2, 0, 2, 2, 0, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 3, 0, 0, 2, 2, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 3, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 0, 2, 2, 1, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 2, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 2, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 2, 2, 0, 0, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57728>, <20, 1028864>, <24, 7993520>, <28, 39292160>, 
            <32, 143611068>, <36, 302515200>, <40, 548039952>, <44, 746732544>, <48, 785948080>, <52, 716001792>, <56, 
            493516768>, <60, 288140800>, <64, 142535366>, <68, 49628160>, <72, 22584096>, <76, 3963904>, <80, 2877344>, 
            <84, 176896>, <88, 295536>, <92, 3328>, <96, 22780>, <104, 1232>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 0, 0, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 0, 0, 2, 2, 2, 0, 0, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 2, 2, 2, 0, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 3, 0, 2, 2, 2, 2, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 1, 2, 2, 2, 2, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 2, 1, 0, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 0, 2, 1, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 2, 0, 3, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58160>, <20, 1020416>, <24, 8030112>, <28, 39276032>, 
            <32, 143480460>, <36, 302667776>, <40, 548155936>, <44, 746498048>, <48, 786022032>, <52, 716041216>, <56, 
            493363008>, <60, 288314368>, <64, 142576742>, <68, 49502208>, <72, 22606144>, <76, 3975168>, <80, 2875920>, 
            <84, 184832>, <88, 291104>, <92, 3584>, <96, 22668>, <104, 1184>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 2, 2, 0, 0, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 2, 0, 2, 0, 2, 2, 0, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 0, 3, 2, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 0, 0, 2, 3, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 2, 0, 2, 3, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 0, 2, 2, 2, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58656>, <20, 1010944>, <24, 8071184>, <28, 39260928>, 
            <32, 143318940>, <36, 302840832>, <40, 548358448>, <44, 746177536>, <48, 785966064>, <52, 716238336>, <56, 
            493322144>, <60, 288330240>, <64, 142559878>, <68, 49462272>, <72, 22649184>, <76, 3965952>, <80, 2865088>, 
            <84, 191744>, <88, 291152>, <92, 4864>, <96, 21596>, <104, 1136>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 0, 0, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 0, 0, 0, 2, 2, 0, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 3, 0, 2, 2, 0, 2, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 3, 2, 2, 0, 0, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 2, 3, 0, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 1, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 2, 0, 3, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 2, 0, 2, 0, 1, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 2, 0, 2, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 2, 0, 0, 0, 2, 2, 2, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57216>, <20, 1029888>, <24, 8002672>, <28, 39291136>, 
            <32, 143564092>, <36, 302494720>, <40, 548155856>, <44, 746818560>, <48, 785790896>, <52, 715844096>, <56, 
            493632352>, <60, 288298496>, <64, 142501318>, <68, 49542144>, <72, 22574752>, <76, 3984384>, <80, 2886560>, 
            <84, 177920>, <88, 293680>, <92, 2304>, <96, 22908>, <104, 1168>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 0, 0, 2, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 0, 2, 2, 2, 2, 2, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 1, 2, 0, 2, 0, 2, 0, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 3, 2, 2, 0, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 3, 2, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 1, 2, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 2, 1, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 2, 0, 2, 1, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 0, 2, 2, 0, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 2, 2, 2, 2, 0, 2, 2, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57808>, <20, 1020160>, <24, 8029568>, <28, 39276288>, 
            <32, 143491948>, <36, 302672896>, <40, 548119744>, <44, 746476544>, <48, 786073296>, <52, 716080640>, <56, 
            493329408>, <60, 288274944>, <64, 142580774>, <68, 49523712>, <72, 22612096>, <76, 3970048>, <80, 2874672>, 
            <84, 184576>, <88, 289920>, <92, 3840>, <96, 23020>, <104, 1216>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 0, 2, 2, 2, 0, 0, 0, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 3, 0, 0, 0, 2, 2, 2, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 2, 0, 0, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 0, 1, 0, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 2, 3, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 3, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 2, 2, 0, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 0, 0, 2, 0, 0, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58448>, <20, 1011456>, <24, 8077312>, <28, 39260416>, 
            <32, 143285484>, <36, 302830592>, <40, 548438592>, <44, 746220544>, <48, 785870800>, <52, 716159488>, <56, 
            493364480>, <60, 288409088>, <64, 142585638>, <68, 49419264>, <72, 22606720>, <76, 3976192>, <80, 2886064>, 
            <84, 192256>, <88, 286976>, <92, 4352>, <96, 21868>, <104, 1088>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 2, 2, 2, 2, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 2, 2, 2, 0, 2, 2, 2, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 1, 2, 0, 0, 0, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 2, 1, 2, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 0, 1, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 2, 2, 2, 3, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 2, 0, 2, 2, 3, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 0, 2, 0, 2, 0, 0, 0, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58544>, <20, 1010688>, <24, 8084512>, <28, 39261184>, 
            <32, 143237900>, <36, 302845952>, <40, 548562848>, <44, 746156032>, <48, 785708432>, <52, 716277760>, <56, 
            493462592>, <60, 288290816>, <64, 142585190>, <68, 49483776>, <72, 22574144>, <76, 3960832>, <80, 2900368>, 
            <84, 191488>, <88, 286624>, <92, 5120>, <96, 21260>, <104, 1056>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 2, 2, 2, 0, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 0, 0, 0, 2, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 3, 2, 2, 0, 2, 2, 0, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 2, 1, 2, 0, 2, 0, 0, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 1, 2, 0, 0, 2, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 1, 2, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 2, 0, 1, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 0, 3, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 0, 0, 2, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 0, 0, 2, 2, 2, 2, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57472>, <20, 1020416>, <24, 8048176>, <28, 39276032>, 
            <32, 143379644>, <36, 302667776>, <40, 548415120>, <44, 746498048>, <48, 785657776>, <52, 716041216>, <56, 
            493648608>, <60, 288314368>, <64, 142466758>, <68, 49502208>, <72, 22614560>, <76, 3975168>, <80, 2879136>, 
            <84, 184832>, <88, 293360>, <92, 3584>, <96, 21756>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 0, 0, 2, 0, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 2, 0, 2, 0, 0, 2, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 3, 2, 0, 2, 0, 0, 2, 2, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 0, 2, 2, 2, 0, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 3, 0, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 1, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 2, 0, 0, 3, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 0, 2, 2, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 0, 2, 2, 0, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58496>, <20, 1011968>, <24, 8081904>, <28, 39259904>, 
            <32, 143254332>, <36, 302820352>, <40, 548524880>, <44, 746263552>, <48, 785743280>, <52, 716080640>, <56, 
            493471328>, <60, 288487936>, <64, 142536134>, <68, 49376256>, <72, 22619552>, <76, 3986432>, <80, 2881696>, 
            <84, 192768>, <88, 289456>, <92, 3840>, <96, 21372>, <104, 1040>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 0, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 0, 2, 2, 2, 2, 2, 0, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 2, 2, 0, 2, 2, 2, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 2, 0, 0, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 3, 2, 2, 0, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 1, 2, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 2, 2, 0, 2, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 2, 2, 0, 2, 2, 2, 0, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 2, 0, 2, 2, 0, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 2, 0, 0, 2, 2, 2, 0, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56816>, <20, 1037824>, <24, 7953568>, <28, 39307776>, 
            <32, 143775052>, <36, 302352384>, <40, 547823264>, <44, 747010048>, <48, 786027536>, <52, 715883520>, <56, 
            493540160>, <60, 288046080>, <64, 142557158>, <68, 49711104>, <72, 22540352>, <76, 3962880>, <80, 2890192>, 
            <84, 169472>, <88, 293920>, <92, 2560>, <96, 24140>, <104, 1312>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 2, 0, 2, 0, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 1, 2, 2, 2, 2, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 2, 3, 2, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 2, 3, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 1, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 0, 2, 0, 0, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 2, 0, 0, 0, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 2, 2, 2, 0, 2, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58128>, <20, 1020160>, <24, 8036544>, <28, 39276288>, 
            <32, 143439404>, <36, 302672896>, <40, 548266880>, <44, 746476544>, <48, 785861456>, <52, 716080640>, <56, 
            493493376>, <60, 288274944>, <64, 142520742>, <68, 49523712>, <72, 22617600>, <76, 3970048>, <80, 2872176>, 
            <84, 184576>, <88, 294080>, <92, 3840>, <96, 21932>, <104, 1152>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 2, 0, 0, 2, 0, 2, 2, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 2, 0, 2, 0, 2, 0, 2, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 3, 2, 0, 2, 2, 0, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 3, 0, 0, 0, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 3, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 2, 2, 1, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 0, 0, 3, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 2, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 2, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57760>, <20, 1020160>, <24, 8022928>, <28, 39276288>, 
            <32, 143535388>, <36, 302672896>, <40, 548004784>, <44, 746476544>, <48, 786229488>, <52, 716080640>, <56, 
            493222560>, <60, 288274944>, <64, 142601606>, <68, 49523712>, <72, 22627936>, <76, 3970048>, <80, 2866752>, 
            <84, 184576>, <88, 289488>, <92, 3840>, <96, 23516>, <104, 1264>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 2, 0, 2, 0, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 3, 0, 0, 2, 2, 0, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 3, 0, 0, 0, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 3, 2, 0, 2, 0, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 3, 0, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 2, 3, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 2, 2, 3, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 2, 0, 2, 0, 2, 2, 2, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 2, 2, 2, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56832>, <20, 1028608>, <24, 8011632>, <28, 39292416>, 
            <32, 143516092>, <36, 302520320>, <40, 548273616>, <44, 746711040>, <48, 785638576>, <52, 716041216>, <56, 
            493729120>, <60, 288101376>, <64, 142492358>, <68, 49649664>, <72, 22549152>, <76, 3958784>, <80, 2900000>, 
            <84, 176640>, <88, 292400>, <92, 3584>, <96, 22524>, <104, 1168>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 2, 0, 2, 0, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 2, 2, 0, 2, 0, 2, 0, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 2, 0, 2, 0, 0, 2, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 0, 0, 0, 0, 0, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 1, 0, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 2, 1, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 2, 3, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 2, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 2, 2, 2, 0, 2, 0, 2, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57408>, <20, 1027840>, <24, 7991664>, <28, 39293184>, 
            <32, 143629820>, <36, 302535680>, <40, 547987024>, <44, 746646528>, <48, 786016560>, <52, 716159488>, <56, 
            493481824>, <60, 287983104>, <64, 142523718>, <68, 49714176>, <72, 22607264>, <76, 3943424>, <80, 2867552>, 
            <84, 175872>, <88, 296496>, <92, 4352>, <96, 22844>, <104, 1296>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 2, 2, 2, 0, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 0, 0, 0, 0, 2, 2, 2, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 2, 2, 0, 0, 2, 0, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 1, 0, 2, 2, 0, 0, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 3, 0, 2, 2, 0, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 3, 2, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 0, 0, 3, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 2, 0, 0, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 2, 2, 0, 0, 0, 2, 2, 0, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58816>, <20, 1012224>, <24, 8068528>, <28, 39259648>, 
            <32, 143332604>, <36, 302815232>, <40, 548323472>, <44, 746285056>, <48, 786017840>, <52, 716041216>, <56, 
            493275104>, <60, 288527360>, <64, 142587206>, <68, 49354752>, <72, 22637600>, <76, 3991552>, <80, 2869984>, 
            <84, 193024>, <88, 289136>, <92, 3584>, <96, 22076>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 0, 2, 2, 0, 0, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 0, 2, 0, 0, 2, 0, 2, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 3, 2, 2, 2, 0, 0, 0, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 3, 2, 0, 2, 0, 0, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 3, 2, 2, 0, 2, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 3, 2, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 1, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 0, 2, 3, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 2, 0, 2, 1, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 0, 2, 0, 2, 0, 2, 2, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 0, 2, 2, 2, 0, 0, 2, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 2, 0, 0, 0, 0, 0, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57648>, <20, 1020160>, <24, 8045408>, <28, 39276288>, 
            <32, 143393548>, <36, 302672896>, <40, 548380384>, <44, 746476544>, <48, 785707152>, <52, 716080640>, <56, 
            493607616>, <60, 288274944>, <64, 142485350>, <68, 49523712>, <72, 22611136>, <76, 3970048>, <80, 2878992>, 
            <84, 184576>, <88, 293344>, <92, 3840>, <96, 21772>, <104, 1120>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 2, 2, 2, 0, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 0, 2, 2, 2, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 3, 2, 2, 2, 0, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 3, 0, 2, 0, 2, 0, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 1, 2, 0, 0, 2, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 2, 1, 2, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 0, 3, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 2, 0, 2, 3, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 2, 2, 2, 1, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 2, 2, 0, 0, 2, 0, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 2, 2, 0, 0, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57104>, <20, 1028608>, <24, 8011968>, <28, 39292416>, 
            <32, 143506988>, <36, 302520320>, <40, 548305792>, <44, 746711040>, <48, 785582928>, <52, 716041216>, <56, 
            493786240>, <60, 288101376>, <64, 142453158>, <68, 49649664>, <72, 22570496>, <76, 3958784>, <80, 2889584>, 
            <84, 176640>, <88, 296128>, <92, 3584>, <96, 21932>, <104, 1152>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 0, 0, 2, 0, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 2, 2, 0, 2, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 3, 2, 2, 0, 0, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 3, 0, 0, 2, 2, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 3, 2, 2, 0, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 3, 2, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 0, 3, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 2, 2, 2, 3, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 2, 2, 0, 3, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 2, 2, 0, 0, 2, 2, 2, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 2, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57520>, <20, 1019904>, <24, 8047712>, <28, 39276544>, 
            <32, 143381644>, <36, 302678016>, <40, 548410080>, <44, 746455040>, <48, 785665936>, <52, 716120064>, <56, 
            493639872>, <60, 288235520>, <64, 142472806>, <68, 49545216>, <72, 22612160>, <76, 3964928>, <80, 2879376>, 
            <84, 184320>, <88, 293600>, <92, 4096>, <96, 21644>, <104, 1120>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 0, 2, 2, 0, 2, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 2, 0, 0, 0, 2, 2, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 3, 2, 0, 0, 0, 2, 2, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 2, 0, 2, 0, 0, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 3, 0, 0, 2, 0, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 3, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 3, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 3, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 2, 2, 2, 0, 0, 0, 2, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 2, 0, 0, 2, 2, 2, 2, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57648>, <20, 1029632>, <24, 7988192>, <28, 39291392>, 
            <32, 143647244>, <36, 302499840>, <40, 547941728>, <44, 746797056>, <48, 786087056>, <52, 715883520>, <56, 
            493411264>, <60, 288259072>, <64, 142571878>, <68, 49563648>, <72, 22582720>, <76, 3979264>, <80, 2877968>, 
            <84, 177664>, <88, 292960>, <92, 2560>, <96, 23564>, <104, 1248>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 0, 2, 2, 0, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 2, 0, 2, 2, 0, 0, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 2, 2, 0, 2, 0, 2, 2, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 2, 2, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 1, 0, 2, 2, 2, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 3, 0, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 0, 2, 1, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 0, 0, 1, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 2, 2, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57280>, <20, 1027840>, <24, 8002096>, <28, 39293184>, 
            <32, 143566332>, <36, 302535680>, <40, 548151056>, <44, 746646528>, <48, 785796656>, <52, 716159488>, <56, 
            493629664>, <60, 287983104>, <64, 142498630>, <68, 49714176>, <72, 22580512>, <76, 3943424>, <80, 2881760>, 
            <84, 175872>, <88, 295920>, <92, 4352>, <96, 22332>, <104, 1232>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 2, 2, 2, 2, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 0, 2, 2, 2, 2, 0, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 2, 2, 2, 0, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 3, 0, 2, 2, 2, 0, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 3, 2, 0, 2, 0, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 1, 0, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 2, 2, 1, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 0, 0, 2, 1, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 2, 2, 0, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 0, 2, 2, 2, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 2, 0, 2, 0, 0, 2, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57280>, <20, 1028608>, <24, 7988848>, <28, 39292416>, 
            <32, 143651708>, <36, 302520320>, <40, 547922512>, <44, 746711040>, <48, 786118704>, <52, 716041216>, <56, 
            493385056>, <60, 288101376>, <64, 142582854>, <68, 49649664>, <72, 22579616>, <76, 3958784>, <80, 2880224>, 
            <84, 176640>, <88, 291632>, <92, 3584>, <96, 23740>, <104, 1296>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 2, 0, 2, 0, 2, 2, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 2, 0, 2, 2, 0, 0, 2, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 1, 0, 2, 0, 2, 2, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 3, 2, 2, 2, 0, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 2, 3, 2, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 0, 3, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 1, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 2, 0, 0, 3, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 2, 2, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 55888>, <20, 1038080>, <24, 7984192>, <28, 39307520>, 
            <32, 143599724>, <36, 302347264>, <40, 548273536>, <44, 747031552>, <48, 785407440>, <52, 715844096>, <56, 
            493998464>, <60, 288085504>, <64, 142416934>, <68, 49689600>, <72, 22517760>, <76, 3968000>, <80, 2910640>, 
            <84, 169728>, <88, 294976>, <92, 2304>, <96, 22764>, <104, 1152>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 3, 2, 0, 2, 2, 2, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 0, 2, 0, 0, 2, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 2, 2, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 0, 3, 0, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 1, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 0, 0, 2, 1, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 3, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 0, 2, 0, 2, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 0, 0, 2, 0, 2, 2, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57216>, <20, 1029120>, <24, 7996592>, <28, 39291904>, 
            <32, 143603388>, <36, 302510080>, <40, 548050192>, <44, 746754048>, <48, 785940912>, <52, 715962368>, <56, 
            493516768>, <60, 288180224>, <64, 142542534>, <68, 49606656>, <72, 22573856>, <76, 3969024>, <80, 2885024>, 
            <84, 177152>, <88, 292464>, <92, 3072>, <96, 23292>, <104, 1232>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 2, 2, 0, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 0, 0, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 2, 2, 2, 0, 2, 0, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 3, 2, 2, 2, 2, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 0, 2, 2, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 3, 2, 2, 0, 0, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 1, 0, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 0, 2, 3, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 0, 0, 1, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 2, 0, 0, 0, 3, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 0, 2, 2, 0, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 2, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57184>, <20, 1037056>, <24, 7923472>, <28, 39308544>, 
            <32, 143958492>, <36, 302367744>, <40, 547348144>, <44, 746945536>, <48, 786667632>, <52, 716001792>, <56, 
            493104032>, <60, 287927808>, <64, 142639366>, <68, 49775616>, <72, 22611552>, <76, 3947520>, <80, 2851072>, 
            <84, 168704>, <88, 295504>, <92, 3328>, <96, 25500>, <104, 1520>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 0, 0, 2, 2, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 2, 2, 0, 0, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 0, 2, 2, 0, 2, 0, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 3, 0, 2, 2, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 1, 2, 0, 0, 2, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 2, 3, 0, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 2, 2, 3, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 0, 2, 1, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 0, 2, 2, 0, 3, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 2, 0, 2, 0, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 2, 2, 2, 0, 0, 0, 2, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57104>, <20, 1029120>, <24, 7990592>, <28, 39291904>, 
            <32, 143643948>, <36, 302510080>, <40, 547942912>, <44, 746754048>, <48, 786083664>, <52, 715962368>, <56, 
            493426048>, <60, 288180224>, <64, 142549926>, <68, 49606656>, <72, 22597376>, <76, 3969024>, <80, 2874224>, 
            <84, 177152>, <88, 292672>, <92, 3072>, <96, 23724>, <104, 1280>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 2, 2, 2, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 0, 0, 2, 2, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 2, 0, 0, 2, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 3, 2, 2, 2, 2, 2, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 3, 2, 2, 2, 2, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 2, 1, 0, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 0, 2, 1, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 0, 2, 3, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 0, 2, 0, 3, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 2, 2, 2, 0, 2, 0, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 2, 2, 0, 0, 2, 2, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56112>, <20, 1037568>, <24, 7979936>, <28, 39308032>, 
            <32, 143621772>, <36, 302357504>, <40, 548219424>, <44, 746988544>, <48, 785479312>, <52, 715922944>, <56, 
            493948736>, <60, 288006656>, <64, 142427238>, <68, 49732608>, <72, 22526272>, <76, 3957760>, <80, 2904592>, 
            <84, 169216>, <88, 296224>, <92, 2816>, <96, 22668>, <104, 1184>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 0, 2, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 2, 0, 2, 2, 0, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 1, 2, 0, 0, 0, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 3, 0, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 2, 2, 3, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 2, 2, 2, 1, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 2, 0, 2, 0, 3, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 2, 0, 0, 2, 2, 2, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58224>, <20, 1020416>, <24, 8020320>, <28, 39276032>, 
            <32, 143541068>, <36, 302667776>, <40, 547999584>, <44, 746498048>, <48, 786228496>, <52, 716041216>, <56, 
            493231296>, <60, 288314368>, <64, 142588390>, <68, 49502208>, <72, 22640576>, <76, 3975168>, <80, 2858832>, 
            <84, 184832>, <88, 292320>, <92, 3584>, <96, 23116>, <104, 1248>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 0, 0, 2, 2, 0, 2, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 2, 0, 0, 0, 2, 2, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 0, 0, 2, 0, 0, 2, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0, 2, 0, 2, 0, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 3, 2, 0, 2, 2, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 3, 2, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 0, 3, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0, 2, 0, 1, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 2, 2, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 2, 0, 0, 0, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 2, 0, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58976>, <20, 1011200>, <24, 8057808>, <28, 39260672>, 
            <32, 143397212>, <36, 302835712>, <40, 548157040>, <44, 746199040>, <48, 786240624>, <52, 716198912>, <56, 
            493125920>, <60, 288369664>, <64, 142610950>, <68, 49440768>, <72, 22667232>, <76, 3971072>, <80, 2853376>, 
            <84, 192000>, <88, 290832>, <92, 4608>, <96, 22300>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 0, 0, 0, 2, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 0, 0, 2, 0, 0, 2, 2, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 2, 0, 2, 2, 0, 0, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, 2, 2, 0, 0, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 3, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 2, 0, 3, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 2, 1, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 2, 2, 0, 2, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 2, 2, 0, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57184>, <20, 1029376>, <24, 7990800>, <28, 39291648>, 
            <32, 143641564>, <36, 302504960>, <40, 547946928>, <44, 746775552>, <48, 786088048>, <52, 715922944>, <56, 
            493402528>, <60, 288219648>, <64, 142585094>, <68, 49585152>, <72, 22570080>, <76, 3974144>, <80, 2885888>, 
            <84, 177408>, <88, 290128>, <92, 2816>, <96, 23964>, <104, 1264>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 0, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 0, 0, 2, 2, 0, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 2, 0, 0, 0, 0, 2, 0, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 2, 2, 2, 0, 0, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 3, 2, 0, 2, 2, 2, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 1, 0, 0, 2, 2, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 3, 2, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 0, 3, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 2, 1, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 2, 2, 2, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 2, 2, 0, 2, 2, 2, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58144>, <20, 1012480>, <24, 8096592>, <28, 39259392>, 
            <32, 143168796>, <36, 302810112>, <40, 548743024>, <44, 746306560>, <48, 785451504>, <52, 716001792>, <56, 
            493668896>, <60, 288566784>, <64, 142500742>, <68, 49333248>, <72, 22584288>, <76, 3996672>, <80, 2901952>, 
            <84, 193280>, <88, 287632>, <92, 3328>, <96, 20956>, <104, 944>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 0, 2, 0, 0, 2, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 2, 3, 0, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 3, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 2, 0, 0, 3, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 3, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 2, 0, 2, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 2, 2, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58144>, <20, 1020160>, <24, 8029264>, <28, 39276288>, 
            <32, 143485724>, <36, 302672896>, <40, 548144240>, <44, 746476544>, <48, 786031088>, <52, 716080640>, <56, 
            493370400>, <60, 288274944>, <64, 142555014>, <68, 49523712>, <72, 22625760>, <76, 3970048>, <80, 2867136>, 
            <84, 184576>, <88, 293008>, <92, 3840>, <96, 22492>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 2, 0, 0, 2, 0, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 0, 2, 2, 2, 2, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 0, 0, 2, 2, 2, 2, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0, 2, 0, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 1, 2, 0, 0, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 2, 1, 0, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 2, 3, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 0, 1, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 0, 0, 2, 3, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 2, 2, 0, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56704>, <20, 1028608>, <24, 8022064>, <28, 39292416>, 
            <32, 143452604>, <36, 302520320>, <40, 548437648>, <44, 746711040>, <48, 785418672>, <52, 716041216>, <56, 
            493876960>, <60, 288101376>, <64, 142467270>, <68, 49649664>, <72, 22522400>, <76, 3958784>, <80, 2914208>, 
            <84, 176640>, <88, 291824>, <92, 3584>, <96, 22012>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 3, 0, 0, 2, 2, 0, 2, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 2, 0, 2, 2, 0, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 2, 2, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 0, 1, 0, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 2, 2, 3, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 2, 3, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 2, 2, 2, 2, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 2, 2, 0, 0, 2, 0, 0, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57376>, <20, 1028864>, <24, 7991952>, <28, 39292160>, 
            <32, 143628700>, <36, 302515200>, <40, 547989424>, <44, 746732544>, <48, 786013680>, <52, 716001792>, <56, 
            493483168>, <60, 288140800>, <64, 142525062>, <68, 49628160>, <72, 22604384>, <76, 3963904>, <80, 2869952>, 
            <84, 176896>, <88, 295376>, <92, 3328>, <96, 23132>, <104, 1264>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 0, 2, 2, 2, 2, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 2, 2, 2, 0, 2, 2, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 0, 2, 0, 2, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 1, 2, 2, 2, 2, 0, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 1, 0, 0, 2, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 2, 0, 3, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 2, 1, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 2, 0, 0, 1, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 2, 2, 2, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58560>, <20, 1011456>, <24, 8074160>, <28, 39260416>, 
            <32, 143302652>, <36, 302830592>, <40, 548397200>, <44, 746220544>, <48, 785921072>, <52, 716159488>, <56, 
            493339616>, <60, 288409088>, <64, 142576454>, <68, 49419264>, <72, 22625312>, <76, 3976192>, <80, 2876896>, 
            <84, 192256>, <88, 288624>, <92, 4352>, <96, 21820>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 1, 0, 2, 0, 2, 2, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 2, 2, 2, 0, 2, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 1, 2, 2, 2, 2, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 2, 3, 0, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 2, 3, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 2, 0, 1, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 0, 2, 2, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 0, 0, 2, 2, 2, 2, 2, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58288>, <20, 1020416>, <24, 8019680>, <28, 39276032>, 
            <32, 143543948>, <36, 302667776>, <40, 547991904>, <44, 746498048>, <48, 786241936>, <52, 716041216>, <56, 
            493215168>, <60, 288314368>, <64, 142601830>, <68, 49502208>, <72, 22632896>, <76, 3975168>, <80, 2861712>, 
            <84, 184832>, <88, 291680>, <92, 3584>, <96, 23180>, <104, 1248>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 0, 2, 0, 2, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0, 2, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 0, 2, 2, 2, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 1, 2, 0, 2, 2, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 3, 0, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 0, 2, 2, 1, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 0, 2, 2, 3, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 2, 2, 0, 0, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 0, 2, 0, 2, 2, 2, 0, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57456>, <20, 1029120>, <24, 7992160>, <28, 39291904>, 
            <32, 143626316>, <36, 302510080>, <40, 547993440>, <44, 746754048>, <48, 786018064>, <52, 715962368>, <56, 
            493459648>, <60, 288180224>, <64, 142560230>, <68, 49606656>, <72, 22577088>, <76, 3969024>, <80, 2881616>, 
            <84, 177152>, <88, 292832>, <92, 3072>, <96, 23372>, <104, 1248>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 0, 2, 0, 0, 2, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 1, 0, 0, 2, 0, 2, 2, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 3, 2, 0, 0, 0, 2, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 1, 2, 0, 2, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 1, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 2, 2, 1, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 1, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 2, 0, 2, 2, 3, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 2, 2, 2, 2, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57536>, <20, 1029888>, <24, 7981168>, <28, 39291136>, 
            <32, 143693948>, <36, 302494720>, <40, 547820112>, <44, 746818560>, <48, 786244144>, <52, 715844096>, <56, 
            493320544>, <60, 288298496>, <64, 142564934>, <68, 49542144>, <72, 22620576>, <76, 3984384>, <80, 2861024>, 
            <84, 177920>, <88, 294192>, <92, 2304>, <96, 23996>, <104, 1296>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 2, 0, 2, 2, 2, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 0, 2, 0, 2, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 2, 2, 0, 0, 2, 2, 2, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 3, 0, 2, 2, 0, 0, 0, 2, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 3, 2, 2, 0, 0, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 3, 0, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 1, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 0, 3, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 0, 2, 2, 3, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 2, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 0, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 2, 2, 0, 2, 2, 2, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57040>, <20, 1029120>, <24, 8019712>, <28, 39291904>, 
            <32, 143458668>, <36, 302510080>, <40, 548433472>, <44, 746754048>, <48, 785405136>, <52, 715962368>, <56, 
            493917952>, <60, 288180224>, <64, 142412838>, <68, 49606656>, <72, 22564736>, <76, 3969024>, <80, 2894384>, 
            <84, 177152>, <88, 296960>, <92, 3072>, <96, 21484>, <104, 1088>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 2, 2, 2, 2, 2, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 2, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 2, 2, 2, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 2, 0, 2, 2, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 2, 1, 0, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 2, 2, 1, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 0, 2, 0, 1, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 0, 2, 0, 2, 1, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 0, 0, 0, 2, 2, 0, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57024>, <20, 1028864>, <24, 8008688>, <28, 39292160>, 
            <32, 143530876>, <36, 302515200>, <40, 548236240>, <44, 746732544>, <48, 785693232>, <52, 716001792>, <56, 
            493680736>, <60, 288140800>, <64, 142518342>, <68, 49628160>, <72, 22540448>, <76, 3963904>, <80, 2902496>, 
            <84, 176896>, <88, 291504>, <92, 3328>, <96, 22716>, <104, 1168>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 0, 2, 0, 2, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 2, 0, 0, 0, 2, 2, 2, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 3, 2, 2, 2, 0, 0, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 2, 2, 0, 2, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 3, 2, 0, 2, 2, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 3, 0, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 1, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 0, 2, 0, 3, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 0, 2, 2, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58800>, <20, 1019392>, <24, 8010528>, <28, 39277056>, 
            <32, 143590924>, <36, 302688256>, <40, 547876000>, <44, 746412032>, <48, 786399120>, <52, 716198912>, <56, 
            493099584>, <60, 288156672>, <64, 142635878>, <68, 49588224>, <72, 22642240>, <76, 3954688>, <80, 2852496>, 
            <84, 183808>, <88, 293536>, <92, 4608>, <96, 23052>, <104, 1312>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 2, 0, 0, 2, 0, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 0, 2, 0, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 2, 2, 0, 2, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 0, 2, 0, 2, 0, 0, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 2, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 1, 2, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 3, 0, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 3, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 0, 2, 2, 1, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 0, 2, 2, 0, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 0, 0, 2, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 0, 0, 2, 0, 2, 0, 2, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57904>, <20, 1020672>, <24, 8040800>, <28, 39275776>, 
            <32, 143417356>, <36, 302662656>, <40, 548320992>, <44, 746519552>, <48, 785789584>, <52, 716001792>, <56, 
            493543104>, <60, 288353792>, <64, 142510438>, <68, 49480704>, <72, 22609088>, <76, 3980288>, <80, 2878224>, 
            <84, 185088>, <88, 292832>, <92, 3328>, <96, 22028>, <104, 1120>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 0, 2, 2, 0, 0, 0, 2, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 3, 2, 2, 0, 0, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 0, 0, 0, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 3, 2, 0, 2, 2, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 2, 3, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 2, 3, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 0, 0, 2, 0, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 2, 0, 2, 0, 2, 2, 2, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 0, 2, 2, 2, 2, 0, 0, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 0, 2, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57920>, <20, 1019904>, <24, 8027440>, <28, 39276544>, 
            <32, 143502972>, <36, 302678016>, <40, 548092688>, <44, 746455040>, <48, 786109232>, <52, 716120064>, <56, 
            493304544>, <60, 288235520>, <64, 142585926>, <68, 49545216>, <72, 22616352>, <76, 3964928>, <80, 2871648>, 
            <84, 184320>, <88, 290544>, <92, 4096>, <96, 22972>, <104, 1232>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 2, 2, 0, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 0, 2, 0, 0, 2, 2, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 3, 0, 2, 2, 0, 2, 2, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 2, 2, 2, 2, 2, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 1, 0, 2, 2, 2, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 3, 0, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 2, 1, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 1, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 1, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 0, 0, 2, 0, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 2, 0, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 2, 2, 0, 2, 0, 2, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57776>, <20, 1020416>, <24, 8052256>, <28, 39276032>, 
            <32, 143347724>, <36, 302667776>, <40, 548499360>, <44, 746498048>, <48, 785555344>, <52, 716041216>, <56, 
            493690944>, <60, 288314368>, <64, 142499686>, <68, 49502208>, <72, 22568000>, <76, 3975168>, <80, 2898576>, 
            <84, 184832>, <88, 291232>, <92, 3584>, <96, 21516>, <104, 1056>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 0, 0, 0, 2, 2, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 2, 2, 0, 2, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 2, 2, 0, 0, 2, 2, 2, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 1, 2, 0, 0, 2, 0, 0, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 0, 2, 0, 2, 0, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 1, 2, 0, 2, 0, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 2, 1, 0, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 1, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 0, 2, 2, 3, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 0, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 2, 2, 0, 0, 2, 2, 0, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58624>, <20, 1011712>, <24, 8072496>, <28, 39260160>, 
            <32, 143311676>, <36, 302825472>, <40, 548375184>, <44, 746242048>, <48, 785948848>, <52, 716120064>, <56, 
            493323488>, <60, 288448512>, <64, 142575558>, <68, 49397760>, <72, 22631968>, <76, 3981312>, <80, 2873632>, 
            <84, 192512>, <88, 289008>, <92, 4096>, <96, 21884>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 0, 2, 2, 0, 2, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 2, 2, 2, 2, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 2, 2, 0, 2, 2, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 1, 2, 2, 0, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 1, 0, 2, 0, 2, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 3, 0, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 3, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 2, 0, 0, 1, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 2, 0, 2, 3, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 0, 0, 2, 2, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 2, 2, 0, 0, 2, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57408>, <20, 1028864>, <24, 7986544>, <28, 39292160>, 
            <32, 143663612>, <36, 302515200>, <40, 547892816>, <44, 746732544>, <48, 786159920>, <52, 716001792>, <56, 
            493352800>, <60, 288140800>, <64, 142595398>, <68, 49628160>, <72, 22578592>, <76, 3963904>, <80, 2879840>, 
            <84, 176896>, <88, 291376>, <92, 3328>, <96, 23868>, <104, 1296>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 2, 2, 2, 0, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 0, 2, 0, 0, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 2, 0, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 2, 1, 2, 0, 0, 2, 2, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 2, 2, 2, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 3, 2, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 0, 0, 3, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 2, 2, 3, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 2, 2, 2, 2, 1, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 2, 2, 0, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 59072>, <20, 1020160>, <24, 7997616>, <28, 39276288>, 
            <32, 143667196>, <36, 302672896>, <40, 547679632>, <44, 746476544>, <48, 786665520>, <52, 716080640>, <56, 
            492912096>, <60, 288274944>, <64, 142680902>, <68, 49523712>, <72, 22662688>, <76, 3970048>, <80, 2840544>, 
            <84, 184576>, <88, 292976>, <92, 3840>, <96, 23868>, <104, 1360>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 0, 2, 2, 0, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 2, 2, 2, 2, 2, 2, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 1, 2, 2, 2, 0, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 3, 0, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 2, 1, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 0, 1, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 0, 2, 2, 3, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 2, 2, 2, 0, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 2, 2, 2, 0, 2, 0, 2, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 0, 2, 2, 0, 0, 0, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58912>, <20, 1010944>, <24, 8059472>, <28, 39260928>, 
            <32, 143388188>, <36, 302840832>, <40, 548179056>, <44, 746177536>, <48, 786212848>, <52, 716238336>, <56, 
            493142048>, <60, 288330240>, <64, 142611846>, <68, 49462272>, <72, 22660576>, <76, 3965952>, <80, 2856640>, 
            <84, 191744>, <88, 290448>, <92, 4864>, <96, 22236>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 2, 0, 0, 0, 2, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 2, 0, 2, 0, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 2, 2, 0, 2, 2, 0, 0, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 2, 2, 0, 2, 0, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 2, 2, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 1, 2, 0, 0, 0, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 2, 1, 2, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 0, 2, 3, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 2, 2, 2, 2, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 0, 2, 0, 0, 2, 2, 2, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57424>, <20, 1028608>, <24, 7990464>, <28, 39292416>, 
            <32, 143636844>, <36, 302520320>, <40, 547970048>, <44, 746711040>, <48, 786036176>, <52, 716041216>, <56, 
            493474432>, <60, 288101376>, <64, 142516774>, <68, 49649664>, <72, 22616320>, <76, 3958784>, <80, 2864048>, 
            <84, 176640>, <88, 296640>, <92, 3584>, <96, 23020>, <104, 1280>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 2, 0, 0, 0, 2, 0, 0, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 1, 2, 0, 0, 2, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 3, 2, 2, 0, 2, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 1, 0, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 2, 0, 1, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 0, 0, 0, 2, 3, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 2, 2, 2, 2, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 0, 0, 2, 2, 2, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58112>, <20, 1019136>, <24, 8029616>, <28, 39277312>, 
            <32, 143483964>, <36, 302693376>, <40, 548149520>, <44, 746390528>, <48, 786020528>, <52, 716238336>, <56, 
            493385184>, <60, 288117248>, <64, 142540230>, <68, 49609728>, <72, 22636320>, <76, 3949568>, <80, 2861856>, 
            <84, 183552>, <88, 294768>, <92, 4864>, <96, 22140>, <104, 1232>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 0, 2, 0, 0, 2, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 2, 2, 2, 2, 0, 0, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 3, 0, 2, 0, 0, 2, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 2, 2, 2, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 3, 2, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 0, 2, 2, 3, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 0, 2, 2, 2, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 2, 2, 2, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 2, 0, 0, 2, 0, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 2, 2, 0, 0, 2, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57680>, <20, 1028608>, <24, 7987904>, <28, 39292416>, 
            <32, 143648364>, <36, 302520320>, <40, 547939328>, <44, 746711040>, <48, 786089936>, <52, 716041216>, <56, 
            493409920>, <60, 288101376>, <64, 142570534>, <68, 49649664>, <72, 22585600>, <76, 3958784>, <80, 2875568>, 
            <84, 176640>, <88, 294080>, <92, 3584>, <96, 23276>, <104, 1280>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 0, 0, 0, 2, 2, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 0, 0, 0, 2, 0, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 0, 2, 2, 2, 2, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 3, 0, 0, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 2, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 1, 2, 2, 0, 2, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 3, 0, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 0, 2, 3, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 1, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 2, 2, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 2, 2, 2, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 2, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 2, 2, 2, 0, 0, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 59536>, <20, 1009664>, <24, 8048192>, <28, 39262208>, 
            <32, 143446188>, <36, 302866432>, <40, 548036096>, <44, 746070016>, <48, 786405968>, <52, 716435456>, <56, 
            493001600>, <60, 288133120>, <64, 142651046>, <68, 49569792>, <72, 22674176>, <76, 3940352>, <80, 2844400>, 
            <84, 190464>, <88, 292928>, <92, 6144>, <96, 22060>, <104, 1280>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 0, 2, 2, 2, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 2, 2, 0, 0, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 2, 2, 2, 0, 0, 2, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 2, 2, 2, 2, 2, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 1, 2, 2, 2, 0, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 2, 3, 2, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 0, 3, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 0, 2, 1, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 2, 2, 0, 2, 3, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 2, 0, 0, 2, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58896>, <20, 1012480>, <24, 8059584>, <28, 39259392>, 
            <32, 143387948>, <36, 302810112>, <40, 548178816>, <44, 746306560>, <48, 786215248>, <52, 716001792>, <56, 
            493136000>, <60, 288566784>, <64, 142620582>, <68, 49333248>, <72, 22652416>, <76, 3996672>, <80, 2861680>, 
            <84, 193280>, <88, 288448>, <92, 3328>, <96, 22700>, <104, 1152>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 2, 2, 2, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 2, 0, 0, 2, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 3, 0, 0, 0, 2, 0, 0, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 2, 0, 1, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 0, 2, 0, 1, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 2, 2, 2, 1, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 2, 2, 2, 0, 0, 0, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58368>, <20, 1019648>, <24, 8025008>, <28, 39276800>, 
            <32, 143507772>, <36, 302683136>, <40, 548090128>, <44, 746433536>, <48, 786102960>, <52, 716159488>, <56, 
            493320672>, <60, 288196096>, <64, 142565318>, <68, 49566720>, <72, 22634272>, <76, 3959808>, <80, 2861088>, 
            <84, 184064>, <88, 294256>, <92, 4352>, <96, 22396>, <104, 1232>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 0, 2, 2, 2, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 2, 0, 0, 0, 0, 2, 0, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 2, 0, 2, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 2, 1, 2, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 0, 1, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 0, 0, 2, 3, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 2, 2, 2, 2, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 2, 2, 0, 2, 2, 0, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57920>, <20, 1020928>, <24, 8041648>, <28, 39275520>, 
            <32, 143412092>, <36, 302657536>, <40, 548332688>, <44, 746541056>, <48, 785780528>, <52, 715962368>, <56, 
            493535712>, <60, 288393216>, <64, 142532166>, <68, 49459200>, <72, 22589472>, <76, 3985408>, <80, 2887008>, 
            <84, 185344>, <88, 290928>, <92, 3072>, <96, 22204>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 2, 0, 0, 0, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 0, 2, 2, 2, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 3, 0, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 1, 2, 2, 2, 2, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 1, 2, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 1, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 2, 0, 1, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 1, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 2, 2, 0, 0, 0, 0, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 0, 0, 2, 0, 0, 2, 0, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58656>, <20, 1019648>, <24, 8027216>, <28, 39276800>, 
            <32, 143487260>, <36, 302683136>, <40, 548148336>, <44, 746433536>, <48, 786023920>, <52, 716159488>, <56, 
            493370400>, <60, 288196096>, <64, 142562182>, <68, 49566720>, <72, 22621664>, <76, 3959808>, <80, 2865600>, 
            <84, 184064>, <88, 295056>, <92, 4352>, <96, 21980>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 2, 2, 2, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 2, 0, 2, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 1, 0, 2, 0, 2, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 3, 2, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 0, 0, 1, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 2, 2, 3, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 3, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 0, 2, 0, 2, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 2, 2, 2, 2, 0, 2, 0, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 2, 2, 2, 2, 0, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58208>, <20, 1011968>, <24, 8088848>, <28, 39259904>, 
            <32, 143217116>, <36, 302820352>, <40, 548615344>, <44, 746263552>, <48, 785629296>, <52, 716080640>, <56, 
            493537184>, <60, 288487936>, <64, 142541062>, <68, 49376256>, <72, 22590048>, <76, 3986432>, <80, 2897152>, 
            <84, 192768>, <88, 286800>, <92, 3840>, <96, 21404>, <104, 1008>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 2, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, 2, 0, 0, 0, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 1, 0, 2, 2, 0, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 2, 3, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 0, 3, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 2, 0, 0, 1, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 2, 2, 0, 0, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 2, 0, 0, 0, 0, 2, 2, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57984>, <20, 1019648>, <24, 8038000>, <28, 39276800>, 
            <32, 143432764>, <36, 302683136>, <40, 548284880>, <44, 746433536>, <48, 785829296>, <52, 716159488>, <56, 
            493533024>, <60, 288196096>, <64, 142486470>, <68, 49566720>, <72, 22638240>, <76, 3959808>, <80, 2863776>, 
            <84, 184064>, <88, 296240>, <92, 4352>, <96, 21628>, <104, 1168>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 0, 2, 0, 2, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 0, 0, 2, 0, 0, 2, 2, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 3, 2, 0, 0, 2, 2, 2, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 2, 1, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 0, 1, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 2, 2, 0, 1, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 2, 2, 2, 2, 0, 0, 0, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 2, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 2, 0, 0, 2, 0, 2, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57296>, <20, 1029120>, <24, 7979520>, <28, 39291904>, 
            <32, 143710316>, <36, 302510080>, <40, 547771200>, <44, 746754048>, <48, 786317008>, <52, 715962368>, <56, 
            493262080>, <60, 288180224>, <64, 142588454>, <68, 49606656>, <72, 22616448>, <76, 3969024>, <80, 2862896>, 
            <84, 177152>, <88, 292608>, <92, 3072>, <96, 24300>, <104, 1344>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 0, 2, 0, 2, 0, 2, 2, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 3, 2, 0, 0, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 3, 2, 2, 2, 2, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 2, 1, 0, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 2, 1, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 2, 3, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 2, 2, 2, 3, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 2, 2, 0, 2, 0, 2, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58000>, <20, 1011712>, <24, 8099072>, <28, 39260160>, 
            <32, 143156012>, <36, 302825472>, <40, 548775360>, <44, 746242048>, <48, 785405008>, <52, 716120064>, <56, 
            493708544>, <60, 288448512>, <64, 142480806>, <68, 49397760>, <72, 22590592>, <76, 3981312>, <80, 2899696>, 
            <84, 192512>, <88, 288768>, <92, 4096>, <96, 20652>, <104, 960>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 0, 2, 2, 0, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 0, 0, 0, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 2, 0, 2, 2, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 1, 0, 2, 2, 2, 2, 2, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 2, 0, 2, 0, 0, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 3, 2, 0, 2, 0, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 2, 3, 0, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 2, 2, 3, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 1, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 2, 2, 2, 0, 2, 0, 2, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 2, 0, 2, 2, 2, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57648>, <20, 1028608>, <24, 7973984>, <28, 39292416>, 
            <32, 143738124>, <36, 302520320>, <40, 547701728>, <44, 746711040>, <48, 786415760>, <52, 716041216>, <56, 
            493180096>, <60, 288101376>, <64, 142625638>, <68, 49649664>, <72, 22609600>, <76, 3958784>, <80, 2862608>, 
            <84, 176640>, <88, 292576>, <92, 3584>, <96, 24332>, <104, 1376>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 0, 2, 0, 0, 2, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 2, 0, 2, 2, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 0, 2, 0, 2, 2, 2, 2, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 3, 0, 0, 2, 0, 2, 0, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 3, 0, 0, 0, 0, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 0, 3, 2, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 3, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 2, 2, 2, 0, 2, 2, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 0, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57728>, <20, 1028864>, <24, 7984368>, <28, 39292160>, 
            <32, 143668796>, <36, 302515200>, <40, 547891280>, <44, 746732544>, <48, 786141104>, <52, 716001792>, <56, 
            493401184>, <60, 288140800>, <64, 142533574>, <68, 49628160>, <72, 22626208>, <76, 3963904>, <80, 2857376>, 
            <84, 176896>, <88, 297392>, <92, 3328>, <96, 23164>, <104, 1296>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 2, 0, 2, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 3, 0, 0, 0, 2, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 2, 2, 2, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 1, 0, 2, 0, 0, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 1, 0, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 1, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 2, 2, 2, 3, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 0, 2, 2, 1, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 2, 2, 0, 2, 0, 2, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 2, 0, 0, 2, 2, 2, 2, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 0, 2, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57776>, <20, 1019904>, <24, 8036000>, <28, 39276544>, 
            <32, 143450892>, <36, 302678016>, <40, 548230688>, <44, 746455040>, <48, 785912720>, <52, 716120064>, <56, 
            493459776>, <60, 288235520>, <64, 142524774>, <68, 49545216>, <72, 22623552>, <76, 3964928>, <80, 2870928>, 
            <84, 184320>, <88, 292896>, <92, 4096>, <96, 22284>, <104, 1184>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 0, 2, 2, 2, 2, 0, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 3, 0, 0, 0, 0, 0, 0, 2, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 1, 2, 0, 2, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 1, 0, 2, 2, 0, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 3, 0, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 1, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 0, 2, 3, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 0, 0, 2, 3, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56832>, <20, 1029632>, <24, 8016688>, <28, 39291392>, 
            <32, 143482940>, <36, 302499840>, <40, 548364944>, <44, 746797056>, <48, 785502896>, <52, 715883520>, <56, 
            493844704>, <60, 288259072>, <64, 142436806>, <68, 49563648>, <72, 22564384>, <76, 3979264>, <80, 2895392>, 
            <84, 177664>, <88, 294640>, <92, 2560>, <96, 22140>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 0, 0, 0, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 2, 2, 2, 2, 2, 0, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 3, 0, 2, 2, 2, 0, 0, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 2, 0, 2, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 1, 2, 2, 0, 0, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 3, 0, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 2, 1, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 2, 0, 2, 1, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 0, 0, 0, 3, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 2, 2, 2, 0, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57904>, <20, 1019392>, <24, 8027616>, <28, 39277056>, 
            <32, 143502092>, <36, 302688256>, <40, 548095328>, <44, 746412032>, <48, 786103952>, <52, 716198912>, <56, 
            493311936>, <60, 288156672>, <64, 142578534>, <68, 49588224>, <72, 22621632>, <76, 3954688>, <80, 2869008>, 
            <84, 183808>, <88, 291424>, <92, 4608>, <96, 22796>, <104, 1248>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 2, 0, 0, 2, 2, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 2, 0, 0, 2, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 0, 2, 0, 2, 2, 0, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 3, 2, 2, 0, 2, 2, 0, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 3, 0, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 3, 0, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 2, 0, 3, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 0, 2, 0, 0, 3, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 2, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57408>, <20, 1028608>, <24, 7997744>, <28, 39292416>, 
            <32, 143590524>, <36, 302520320>, <40, 548092688>, <44, 746711040>, <48, 785866544>, <52, 716041216>, <56, 
            493597408>, <60, 288101376>, <64, 142482502>, <68, 49649664>, <72, 22608160>, <76, 3958784>, <80, 2869088>, 
            <84, 176640>, <88, 297712>, <92, 3584>, <96, 22460>, <104, 1232>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 2, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 1, 0, 2, 0, 0, 0, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 3, 0, 0, 0, 0, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 0, 3, 0, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 3, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 2, 0, 1, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 2, 2, 0, 0, 0, 0, 0, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57888>, <20, 1019904>, <24, 8042000>, <28, 39276544>, 
            <32, 143410332>, <36, 302678016>, <40, 548337968>, <44, 746455040>, <48, 785769968>, <52, 716120064>, <56, 
            493550496>, <60, 288235520>, <64, 142517382>, <68, 49545216>, <72, 22600032>, <76, 3964928>, <80, 2881728>, 
            <84, 184320>, <88, 292688>, <92, 4096>, <96, 21852>, <104, 1136>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 2, 2, 0, 2, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 0, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 2, 0, 0, 2, 0, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 2, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 2, 2, 0, 3, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 0, 2, 0, 0, 3, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 2, 2, 2, 0, 2, 0, 0, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 2, 2, 2, 2, 0, 0, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58480>, <20, 1011456>, <24, 8082080>, <28, 39260416>, 
            <32, 143253452>, <36, 302830592>, <40, 548527520>, <44, 746220544>, <48, 785738000>, <52, 716159488>, <56, 
            493478720>, <60, 288409088>, <64, 142528742>, <68, 49419264>, <72, 22624832>, <76, 3976192>, <80, 2879056>, 
            <84, 192256>, <88, 290336>, <92, 4352>, <96, 21196>, <104, 1056>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 2, 0, 0, 2, 0, 0, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 0, 2, 2, 0, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 0, 0, 2, 2, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 3, 2, 2, 0, 2, 0, 0, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 3, 2, 0, 0, 0, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 3, 2, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 2, 0, 1, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 2, 0, 1, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 1, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 2, 0, 2, 2, 2, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58432>, <20, 1011712>, <24, 8074416>, <28, 39260160>, 
            <32, 143303036>, <36, 302825472>, <40, 548398224>, <44, 746242048>, <48, 785908528>, <52, 716120064>, <56, 
            493371872>, <60, 288448512>, <64, 142535238>, <68, 49397760>, <72, 22655008>, <76, 3981312>, <80, 2864992>, 
            <84, 192512>, <88, 290928>, <92, 4096>, <96, 21692>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 0, 2, 2, 0, 2, 0, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 3, 2, 0, 2, 2, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 2, 2, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 3, 0, 0, 2, 2, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 3, 2, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 0, 1, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 3, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 0, 2, 1, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 2, 0, 0, 0, 2, 2, 2, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 2, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57024>, <20, 1029120>, <24, 7998512>, <28, 39291904>, 
            <32, 143594748>, <36, 302510080>, <40, 548073232>, <44, 746754048>, <48, 785900592>, <52, 715962368>, <56, 
            493565152>, <60, 288180224>, <64, 142502214>, <68, 49606656>, <72, 22596896>, <76, 3969024>, <80, 2876384>, 
            <84, 177152>, <88, 294384>, <92, 3072>, <96, 23100>, <104, 1232>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 2, 0, 2, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 2, 2, 2, 0, 0, 2, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 2, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 1, 0, 2, 2, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 3, 0, 0, 0, 0, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 3, 2, 0, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 2, 0, 3, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 2, 2, 2, 3, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 3, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 2, 0, 2, 2, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 2, 2, 2, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57904>, <20, 1020160>, <24, 8033696>, <28, 39276288>, 
            <32, 143462796>, <36, 302672896>, <40, 548200992>, <44, 746476544>, <48, 785953936>, <52, 716080640>, <56, 
            493427520>, <60, 288274944>, <64, 142537318>, <68, 49523712>, <72, 22622528>, <76, 3970048>, <80, 2870544>, 
            <84, 184576>, <88, 292640>, <92, 3840>, <96, 22412>, <104, 1184>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 2, 2, 0, 0, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 0, 2, 2, 0, 2, 0, 0, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 3, 0, 0, 0, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 3, 2, 2, 0, 0, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 1, 2, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 0, 0, 1, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 2, 0, 1, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 1, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 0, 2, 2, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 0, 2, 0, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57440>, <20, 1020160>, <24, 8054608>, <28, 39276288>, 
            <32, 143338588>, <36, 302672896>, <40, 548526064>, <44, 746476544>, <48, 785497200>, <52, 716080640>, <56, 
            493778976>, <60, 288274944>, <64, 142410758>, <68, 49523712>, <72, 22626016>, <76, 3970048>, <80, 2875392>, 
            <84, 184576>, <88, 296336>, <92, 3840>, <96, 21020>, <104, 1072>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 0, 0, 2, 2, 2, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 2, 0, 0, 0, 0, 2, 2, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 2, 0, 0, 2, 0, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 1, 2, 0, 0, 0, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 1, 2, 0, 0, 2, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 2, 1, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 0, 2, 3, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 2, 2, 0, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 2, 2, 0, 2, 2, 0, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58624>, <20, 1011968>, <24, 8071472>, <28, 39259904>, 
            <32, 143317820>, <36, 302820352>, <40, 548360848>, <44, 746263552>, <48, 785963184>, <52, 716080640>, <56, 
            493323488>, <60, 288487936>, <64, 142561222>, <68, 49376256>, <72, 22646304>, <76, 3986432>, <80, 2867488>, 
            <84, 192768>, <88, 290032>, <92, 3840>, <96, 21884>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 2, 2, 0, 2, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 2, 2, 0, 2, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 2, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 3, 2, 2, 0, 2, 0, 2, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 1, 2, 0, 0, 2, 2, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 1, 0, 0, 2, 2, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 3, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 0, 2, 2, 3, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 1, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 2, 2, 2, 2, 0, 0, 0, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 2, 0, 2, 2, 2, 0, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58016>, <20, 1019904>, <24, 8031568>, <28, 39276544>, 
            <32, 143473820>, <36, 302678016>, <40, 548173936>, <44, 746455040>, <48, 785989872>, <52, 716120064>, <56, 
            493402656>, <60, 288235520>, <64, 142542470>, <68, 49545216>, <72, 22626784>, <76, 3964928>, <80, 2867520>, 
            <84, 184320>, <88, 293264>, <92, 4096>, <96, 22364>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 0, 2, 0, 0, 2, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 0, 0, 2, 2, 0, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 0, 0, 2, 2, 0, 2, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 3, 0, 2, 0, 2, 0, 0, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 2, 2, 2, 0, 0, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 3, 2, 0, 0, 0, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 2, 3, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 2, 2, 1, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 0, 1, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 2, 2, 2, 2, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 2, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 0, 2, 2, 2, 0, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 2, 2, 0, 2, 2, 2, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58688>, <20, 1037312>, <24, 7866736>, <28, 39308288>, 
            <32, 144287484>, <36, 302362624>, <40, 546502736>, <44, 746967040>, <48, 787823408>, <52, 715962368>, <56, 
            492269408>, <60, 287967232>, <64, 142868294>, <68, 49754112>, <72, 22677920>, <76, 3952640>, <80, 2804320>, 
            <84, 168960>, <88, 294448>, <92, 3072>, <96, 28220>, <104, 1808>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 0, 2, 2, 2, 2, 2, 0, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 3, 2, 2, 0, 0, 0, 2, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 2, 0, 2, 2, 2, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 1, 2, 2, 0, 2, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 3, 0, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 0, 2, 3, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 2, 0, 2, 3, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 0, 0, 2, 2, 3, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 2, 2, 0, 0, 0, 2, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 2, 0, 2, 2, 2, 0, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57632>, <20, 1028864>, <24, 7980240>, <28, 39292160>, 
            <32, 143697948>, <36, 302515200>, <40, 547810032>, <44, 746732544>, <48, 786260464>, <52, 716001792>, <56, 
            493303072>, <60, 288140800>, <64, 142577030>, <68, 49628160>, <72, 22615776>, <76, 3963904>, <80, 2861504>, 
            <84, 176896>, <88, 294672>, <92, 3328>, <96, 23772>, <104, 1328>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 0, 2, 0, 0, 0, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 0, 2, 2, 2, 2, 0, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 3, 2, 2, 2, 0, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 2, 3, 0, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 2, 3, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 2, 3, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 0, 0, 2, 1, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 2, 0, 0, 2, 2, 2, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58352>, <20, 1020928>, <24, 8026144>, <28, 39275520>, 
            <32, 143501388>, <36, 302657536>, <40, 548104224>, <44, 746541056>, <48, 786091024>, <52, 715962368>, <56, 
            493314624>, <60, 288393216>, <64, 142588390>, <68, 49459200>, <72, 22611776>, <76, 3985408>, <80, 2872272>, 
            <84, 185344>, <88, 291232>, <92, 3072>, <96, 22860>, <104, 1184>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 0, 2, 2, 2, 0, 0, 2, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 2, 2, 0, 2, 2, 2, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 2, 1, 0, 2, 2, 2, 0, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 3, 2, 0, 0, 0, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 2, 1, 2, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 2, 0, 1, 2, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 2, 2, 0, 1, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 1, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 0, 0, 0, 2, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 0, 2, 2, 0, 2, 2, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57728>, <20, 1020928>, <24, 8043568>, <28, 39275520>, 
            <32, 143403452>, <36, 302657536>, <40, 548355728>, <44, 746541056>, <48, 785740208>, <52, 715962368>, <56, 
            493584096>, <60, 288393216>, <64, 142491846>, <68, 49459200>, <72, 22612512>, <76, 3985408>, <80, 2878368>, 
            <84, 185344>, <88, 292848>, <92, 3072>, <96, 22012>, <104, 1104>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 0, 0, 2, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 3, 0, 0, 2, 2, 2, 2, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 2, 2, 0, 2, 2, 2, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 1, 0, 2, 2, 0, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 3, 2, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 3, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 0, 0, 1, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 2, 2, 2, 0, 1, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 2, 2, 2, 0, 0, 2, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 2, 2, 2, 0, 0, 2, 2, 0, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 2, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58384>, <20, 1020928>, <24, 8030912>, <28, 39275520>, 
            <32, 143469356>, <36, 302657536>, <40, 548193152>, <44, 746541056>, <48, 785958224>, <52, 715962368>, <56, 
            493428864>, <60, 288393216>, <64, 142531494>, <68, 49459200>, <72, 22629888>, <76, 3985408>, <80, 2865264>, 
            <84, 185344>, <88, 294592>, <92, 3072>, <96, 22188>, <104, 1152>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 2, 2, 0, 2, 2, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 0, 2, 0, 2, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 0, 0, 0, 2, 2, 0, 2, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 2, 2, 2, 2, 0, 0, 0, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 2, 0, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 1, 0, 0, 2, 0, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 1, 2, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 2, 0, 1, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 0, 3, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 2, 0, 2, 2, 3, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 2, 2, 2, 2, 0, 2, 2, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57360>, <20, 1029120>, <24, 7998208>, <28, 39291904>, 
            <32, 143588524>, <36, 302510080>, <40, 548097728>, <44, 746754048>, <48, 785858384>, <52, 715962368>, <56, 
            493606144>, <60, 288180224>, <64, 142476454>, <68, 49606656>, <72, 22610560>, <76, 3969024>, <80, 2868848>, 
            <84, 177152>, <88, 297472>, <92, 3072>, <96, 22572>, <104, 1216>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 0, 0, 0, 2, 0, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 2, 2, 0, 0, 2, 2, 0, 2, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 3, 0, 2, 2, 0, 0, 2, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 3, 2, 0, 2, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 3, 2, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 3, 2, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 2, 0, 1, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 0, 1, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 2, 0, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 2, 2, 2, 0, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 2, 0, 2, 2, 2, 0, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57184>, <20, 1028608>, <24, 8004048>, <28, 39292416>, 
            <32, 143556188>, <36, 302520320>, <40, 548175472>, <44, 746711040>, <48, 785766000>, <52, 716041216>, <56, 
            493647136>, <60, 288101376>, <64, 142500870>, <68, 49649664>, <72, 22570976>, <76, 3958784>, <80, 2887424>, 
            <84, 176640>, <88, 294416>, <92, 3584>, <96, 22556>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 0, 0, 0, 2, 0, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 2, 2, 0, 0, 2, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 3, 2, 0, 0, 2, 2, 2, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 2, 2, 0, 2, 0, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 1, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 3, 2, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 0, 1, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 2, 2, 2, 3, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 0, 2, 3, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 2, 3, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 0, 2, 2, 0, 2, 0, 2, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 2, 2, 2, 0, 2, 0, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57264>, <20, 1029120>, <24, 8003232>, <28, 39291904>, 
            <32, 143559948>, <36, 302510080>, <40, 548165152>, <44, 746754048>, <48, 785784720>, <52, 715962368>, <56, 
            493623616>, <60, 288180224>, <64, 142521702>, <68, 49606656>, <72, 22558016>, <76, 3969024>, <80, 2892944>, 
            <84, 177152>, <88, 292896>, <92, 3072>, <96, 22796>, <104, 1184>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 0, 0, 2, 2, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 0, 0, 2, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 0, 0, 2, 0, 2, 2, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 3, 0, 0, 0, 0, 0, 2, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 2, 0, 2, 0, 0, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 1, 2, 0, 0, 0, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 1, 2, 2, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 0, 1, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 2, 0, 2, 1, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 3, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 2, 2, 0, 2, 0, 0, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 2, 2, 0, 0, 2, 2, 0, 2, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 2, 2, 0, 0, 0, 2, 0, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57568>, <20, 1019648>, <24, 8055376>, <28, 39276800>, 
            <32, 143332060>, <36, 302683136>, <40, 548539376>, <44, 746433536>, <48, 785495408>, <52, 716159488>, <56, 
            493746720>, <60, 288196096>, <64, 142466310>, <68, 49566720>, <72, 22581984>, <76, 3959808>, <80, 2893440>, 
            <84, 184064>, <88, 293008>, <92, 4352>, <96, 21148>, <104, 1072>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 2, 2, 0, 0, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 0, 0, 0, 0, 0, 2, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 2, 2, 2, 0, 0, 0, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 2, 2, 0, 0, 2, 0, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 2, 3, 0, 2, 2, 0, 2, 2, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 3, 0, 0, 0, 2, 2, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 3, 0, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 2, 2, 1, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 0, 2, 1, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 2, 2, 0, 3, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 2, 2, 0, 2, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 0, 2, 3, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57952>, <20, 1020160>, <24, 8031184>, <28, 39276288>, 
            <32, 143477084>, <36, 302672896>, <40, 548167280>, <44, 746476544>, <48, 785990768>, <52, 716080640>, <56, 
            493418784>, <60, 288274944>, <64, 142514694>, <68, 49523712>, <72, 22648800>, <76, 3970048>, <80, 2858496>, 
            <84, 184576>, <88, 294928>, <92, 3840>, <96, 22300>, <104, 1200>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 0, 0, 2, 0, 0, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 2, 2, 0, 0, 2, 0, 2, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 2, 2, 0, 0, 2, 2, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 2, 0, 0, 2, 2, 0, 2, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 1, 0, 2, 2, 0, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 3, 2, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 0, 0, 2, 1, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 2, 2, 2, 0, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 0, 0, 0, 2, 0, 2, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56464>, <20, 1037824>, <24, 7952000>, <28, 39307776>, 
            <32, 143792684>, <36, 302352384>, <40, 547772736>, <44, 747010048>, <48, 786093136>, <52, 715883520>, <56, 
            493506560>, <60, 288046080>, <64, 142546854>, <68, 49711104>, <72, 22560640>, <76, 3962880>, <80, 2882800>, 
            <84, 169472>, <88, 293760>, <92, 2560>, <96, 24492>, <104, 1344>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 0, 2, 0, 2, 2, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 0, 2, 2, 2, 2, 0, 2, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 0, 0, 0, 0, 2, 2, 2, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 1, 2, 0, 0, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 3, 2, 2, 0, 2, 0, 2, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 1, 0, 0, 0, 2, 2, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 3, 2, 2, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 2, 0, 3, 2, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 0, 0, 1, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 2, 2, 2, 1, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 2, 2, 2, 2, 1, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57840>, <20, 1020672>, <24, 8041440>, <28, 39275776>, 
            <32, 143414476>, <36, 302662656>, <40, 548328672>, <44, 746519552>, <48, 785776144>, <52, 716001792>, <56, 
            493559232>, <60, 288353792>, <64, 142496998>, <68, 49480704>, <72, 22616768>, <76, 3980288>, <80, 2875344>, 
            <84, 185088>, <88, 293472>, <92, 3328>, <96, 21964>, <104, 1120>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 0, 2, 2, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 2, 2, 2, 2, 2, 0, 2, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 3, 2, 2, 0, 2, 2, 2, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 3, 2, 2, 0, 0, 2, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 2, 3, 0, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 3, 0, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 3, 2, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 0, 2, 0, 1, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 1, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 2, 2, 0, 0, 0, 0, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58224>, <20, 1020672>, <24, 8019296>, <28, 39275776>, 
            <32, 143547212>, <36, 302662656>, <40, 547985248>, <44, 746519552>, <48, 786242832>, <52, 716001792>, <56, 
            493231296>, <60, 288353792>, <64, 142574054>, <68, 49480704>, <72, 22654912>, <76, 3980288>, <80, 2852688>, 
            <84, 185088>, <88, 293344>, <92, 3328>, <96, 23116>, <104, 1248>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 0, 0, 0, 0, 2, 2, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 0, 2, 2, 2, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 2, 2, 0, 2, 2, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 3, 2, 0, 2, 0, 0, 0, 0, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 2, 0, 0, 0, 2, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 3, 0, 0, 2, 2, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 3, 0, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 3, 0, 0, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 2, 1, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 2, 2, 1, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 2, 0, 0, 2, 0, 2, 3, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 2, 2, 0, 2, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58160>, <20, 1019648>, <24, 8014880>, <28, 39276800>, 
            <32, 143577484>, <36, 302683136>, <40, 547901600>, <44, 746433536>, <48, 786365072>, <52, 716159488>, <56, 
            493131840>, <60, 288196096>, <64, 142616166>, <68, 49566720>, <72, 22647360>, <76, 3959808>, <80, 2854416>, 
            <84, 184064>, <88, 291744>, <92, 4352>, <96, 23436>, <104, 1312>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 0, 2, 2, 2, 0, 0, 0, 0, 2, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 2, 2, 2, 0, 0, 2, 0, 0, 0, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 0, 2, 2, 0, 0, 2, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 3, 0, 2, 0, 2, 0, 2, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 3, 2, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 1, 0, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 2, 1, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 2, 0, 2, 2, 3, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 2, 0, 0, 0, 2, 0, 2, 3, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 2, 2, 2, 2, 0, 0, 2, 1, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 2, 2, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 2, 2, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57968>, <20, 1020160>, <24, 8033056>, <28, 39276288>, 
            <32, 143465676>, <36, 302672896>, <40, 548193312>, <44, 746476544>, <48, 785967376>, <52, 716080640>, <56, 
            493411392>, <60, 288274944>, <64, 142550758>, <68, 49523712>, <72, 22614848>, <76, 3970048>, <80, 2873424>, 
            <84, 184576>, <88, 292000>, <92, 3840>, <96, 22476>, <104, 1184>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 0, 0, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 0, 2, 0, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 0, 2, 2, 0, 2, 0, 2, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 3, 0, 0, 2, 2, 0, 2, 2, 2, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 2, 2, 0, 0, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 3, 0, 0, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 2, 1, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 2, 2, 2, 1, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 0, 0, 0, 2, 0, 0, 2, 2, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 2, 2, 2, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 2, 0, 2, 2, 0, 2, 2, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 57152>, <20, 1029120>, <24, 7997232>, <28, 39291904>, 
            <32, 143600508>, <36, 302510080>, <40, 548057872>, <44, 746754048>, <48, 785927472>, <52, 715962368>, <56, 
            493532896>, <60, 288180224>, <64, 142529094>, <68, 49606656>, <72, 22581536>, <76, 3969024>, <80, 2882144>, 
            <84, 177152>, <88, 293104>, <92, 3072>, <96, 23228>, <104, 1232>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 0, 2, 0, 0, 2, 2, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 0, 2, 0, 0, 2, 2, 0, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, 2, 2, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 2, 1, 2, 0, 2, 0, 2, 0, 2, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 2, 3, 0, 2, 0, 2, 2, 0, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 2, 0, 2, 3, 0, 0, 0, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 2, 2, 3, 2, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 2, 2, 0, 3, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 2, 2, 2, 0, 0, 2, 2, 2, 0, 3, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 0, 0, 2, 0, 1, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 2, 0, 2, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56640>, <20, 1045760>, <24, 7889008>, <28, 39324416>, 
            <32, 144078076>, <36, 302210048>, <40, 547259216>, <44, 747201536>, <48, 786557744>, <52, 715922944>, <56, 
            493282656>, <60, 287793664>, <64, 142592838>, <68, 49880064>, <72, 22585248>, <76, 3941376>, <80, 2855520>, 
            <84, 161024>, <88, 299312>, <92, 2816>, <96, 25660>, <104, 1552>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 2, 0, 2, 2, 2, 2, 0, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 2, 2, 2, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 2, 2, 2, 2, 0, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 0, 2, 2 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 3, 0, 2, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 2, 1, 0, 2, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 2, 3, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 2, 0, 2, 3, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 3, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 2, 2, 3, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 2, 2, 0, 0, 0, 0, 2, 2, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56944>, <20, 1028864>, <24, 8016608>, <28, 39292160>, 
            <32, 143481676>, <36, 302515200>, <40, 548366560>, <44, 746732544>, <48, 785510160>, <52, 716001792>, <56, 
            493819840>, <60, 288140800>, <64, 142470630>, <68, 49628160>, <72, 22539968>, <76, 3963904>, <80, 2904656>, 
            <84, 176896>, <88, 293216>, <92, 3328>, <96, 22092>, <104, 1120>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 0, 0, 0, 2, 2, 2, 2, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 2, 2, 2, 2, 2, 0, 2, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 3, 0, 2, 0, 0, 2, 0, 0, 0, 2, 2 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, 2, 0, 2, 2, 2, 0, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 2, 2, 2, 0, 0, 0, 0, 2 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 3, 2, 2, 0, 0, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, 2, 1, 2, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 1, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 0, 3, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 2, 2, 0, 1, 0, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 0, 2, 0, 2, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 1 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 56736>, <20, 1037568>, <24, 7953360>, <28, 39308032>, 
            <32, 143777436>, <36, 302357504>, <40, 547819248>, <44, 746988544>, <48, 786023152>, <52, 715922944>, <56, 
            493563680>, <60, 288006656>, <64, 142521990>, <68, 49732608>, <72, 22567648>, <76, 3957760>, <80, 2878528>, 
            <84, 169216>, <88, 296464>, <92, 2816>, <96, 23900>, <104, 1328>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 3, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 0, 0, 0, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 0, 2, 2, 0, 0, 2, 0, 2, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 3, 0, 0, 2, 0, 0, 0, 2, 0, 0 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 2, 0, 2, 0, 0, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 0, 2, 3, 2, 2, 0, 2, 0, 0, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 2, 0, 2, 3, 2, 2, 2, 0, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 0, 0, 3, 0, 2, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 0, 2, 0, 2, 3, 0, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 2, 0, 2, 0, 0, 0, 0, 2, 3, 2, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 0, 2, 0, 1, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 0, 2, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 58560>, <20, 1010688>, <24, 8068080>, <28, 39261184>, 
            <32, 143341948>, <36, 302845952>, <40, 548291536>, <44, 746156032>, <48, 786071088>, <52, 716277760>, <56, 
            493224032>, <60, 288290816>, <64, 142617670>, <68, 49483776>, <72, 22624416>, <76, 3960832>, <80, 2875360>, 
            <84, 191488>, <88, 287408>, <92, 5120>, <96, 22204>, <104, 1168>, <112, 176>, <128, 1> ]>,
    rec<RF1 | 
        mat := [ PowerSequence(IntegerRing()) |
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 2, 0 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 2 ],
            \[ 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 2, 2, 0, 0, 2, 0, 2, 0, 0, 0 ],
            \[ 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 3, 0, 2, 2, 0, 0, 0, 0, 2, 2 ],
            \[ 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 2, 3, 0, 0, 0, 2, 0, 2, 0 ],
            \[ 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 1, 2, 0, 2, 2, 0, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 2, 2, 2, 3, 0, 2, 2, 0 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 3, 2, 0, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 2, 0, 2, 0, 2, 0, 0, 1, 2, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 1, 2 ],
            \[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 0, 0, 2, 2, 0, 2, 0, 0, 0, 3 ],
            \[ 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
            \[ 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
        ],
        dtb := [ car<IntegerRing(), IntegerRing()> | <0, 1>, <16, 55664>, <20, 1037824>, <24, 8001696>, <28, 39307776>, 
            <32, 143492300>, <36, 302352384>, <40, 548556192>, <44, 747010048>, <48, 785013520>, <52, 715883520>, <56, 
            494292800>, <60, 288046080>, <64, 142322406>, <68, 49711104>, <72, 22510144>, <76, 3962880>, <80, 2918224>, 
            <84, 169472>, <88, 298016>, <92, 2560>, <96, 21452>, <104, 1056>, <112, 176>, <128, 1> ]>
]
;