#!/usr/bin/env python3
tab={(1,1):(0,-1),(2,2):(0,-1),(3,3):(0,-1),(1,2):(3,1),(2,1):(3,-1),(2,3):(1,1),(3,2):(1,-1),(3,1):(2,1),(1,3):(2,-1)}
def mul(a,b):
    if a[0]==0: return (b[0],a[1]*b[1])
    if b[0]==0: return (a[0],a[1]*b[1])
    k,s=tab[(a[0],b[0])]
    return (k,a[1]*b[1]*s)
letters=[((1,1),2,1),((1,-1),-2,-1),((2,1),-1,2),((2,-1),1,-2)]
inv=[1,0,3,2]
dp={}
for e,(q,a,b) in enumerate(letters):
    dp[(e,e,q,a,b)]=1
for pos in range(1,12):
    nd={}
    for (f,l,q,x,y),c in dp.items():
        for g,(qq,aa,bb) in enumerate(letters):
            if g==inv[l]: continue
            key=(f,g,mul(q,qq),x+aa,y+bb)
            nd[key]=nd.get(key,0)+c
    dp=nd
res={}
for (f,l,q,x,y),c in dp.items():
    if l==inv[f]: continue
    if q!=(0,-1): continue
    if (x+2*y)%5: continue
    I=sum(1 for e in () )
    e=(abs(2*x-y)+abs(x+2*y)+abs(-3*x-y))//2
    # recover I from counts not possible here
    res[(x,y)]=res.get((x,y),0)+e*c
print("byxy",res)