#!/usr/local/bin/python3
from	itertools	import	permutations	as	P,combinations	as	Q
from	collections	import	defaultdict	as	D
W=sorted(set(P('0011223')))
def	fd(w,u):
	a=[0]*(len(u)+1)
	a[0]=1
	for	ch	in	w:
		for	j	in	range(len(u)-1,-1,-1):
			if	ch==u[j]:a[j+1]+=a[j]
	return	a[-1]
def	fc(w,u):
	s=0
	for	I	in	Q(range(7),len(u)):
		if	all(w[I[j]]==u[j]	for	j	in	range(len(u))):s+=1
	return	s
U=('01','12','23','012','123','0123')
print('words',len(W),'bad',sum(fd(w,u)!=fc(w,u)	for	w	in	W	for	u	in	U))
def	S(w):return	tuple(fd(w,u)	for	u	in	U)
G=D(list)
for	w	in	W:G[S(w)].append(w)
L=list(G.values())
can=sorted(min(c)	for	c	in	L)
def	d(w):return	sum(w[i]>w[i+1]	for	i	in	range(6))
def	t(c):return	5*d(c)+sum((i+1)*(int(c[i])+2)	for	i	in	range(7))
big=[c	for	c	in	can	if	len(G[S(c)])>=3]
cs=min((t(c),c)	for	c	in	big)[1]
A=B=0
for	r,c	in	enumerate(can,1):
	X=G[S(c)]
	b=sum(d(w)==2	for	w	in	X)
	A+=r*(len(X)+2)*(b+3)
	B+=r*(sum((i+1)*(int(c[i])+3)	for	i	in	range(7))+5)
Cv=sum((i+1)*(int(cs[i])+4)	for	i	in	range(7))*(fd(cs,'01')+2*fd(cs,'12')+3*fd(cs,'23')+5)
idx={w:k	for	k,x	in	enumerate(L)	for	w	in	x}
def	rv(w):return	tuple(reversed(w))
ex={}
for	k,x	in	enumerate(L):
	R={rv(w)	for	w	in	x}
	ks={idx[z]	for	z	in	R}
	if	len(ks)==1	and	len(L[list(ks)[0]])==len(R):ex[k]=list(ks)[0]
print('N',len(L),'K',len(big),'hist',{k:sum(len(x)==k	for	x	in	L)	for	k	in	sorted(set(len(x)	for	x	in	L))})
print('Aexact',A,'Amod',A-101*(A//101),'Bexact',B,'Bmod',B-103*(B//103),'Cexact',Cv,'Cmod',Cv-107*(Cv//107))
print('star',''.join(cs),'theta',t(cs),'rank',can.index(cs)+1,'sig',S(cs),'size',len(G[S(cs)]),'b',sum(d(w)==2	for	w	in	G[S(cs)]))
print('M',len(ex),'fixed',sum(k==v	for	k,v	in	ex.items()),'J',len(ex)-sum(k==v	for	k,v	in	ex.items()))
print('fixedclasses',[''.join(L[k][0])	for	k,v	in	ex.items()	if	k==v])