/*
Finding yellow words at distance 7 from the identity.
These words all agree with Identity on positions 8,9,10,11,12.
Note that the (5^7 = 78125) words with |w| = 5 are ignored.
This program does check (implicitly) that the final words found are yellow.
*/

G:=PermutationGroup<12|(1,2,3,4,5,6,7,8,9,10,11),(3,4)(2,10)(6,7)(5,9),(11,12)(1,10)(2,5)(3,7)(4,8)(6,9)>;
M12:=G;
// G = M12.
B:=Base(G); // Not used below.
hamdist:=func<seq1,seq2|#[i:i in [1..#seq1]|seq1[i] ne seq2[i]]>; // Not used either.

SSC:=[];

S5:=[[1..12],[1..12],[1..12],[1..12],[1..12],[1..12],[1..12],[8],[9],[10],[11],[12]];
for i in [1..7] do S5[i]:=[u:u in S5[i]|u ne i];end for;
SS5:=[S5];

SS6:=[];
for i in [1..7] do for j in [u:u in S5[i]|u le 7] do ss:=S5;ss[i]:=[j];Append(~SS6,ss);end for;end for;
// {{#u:u in ss}:ss in SS6};

OSS:=SS6;SS6:=[];
for uu in OSS do pp:=[i:i in [1..12]|#uu[i] eq 1];npp:=#pp;ss:=uu;
for i1 in [1..npp],i2 in [i1+1..npp],i3 in [i2+1..npp],i4 in [i3+1..npp],i5 in [i4+1..npp] do
ps:=[pp[u]:u in [i1,i2,i3,i4,i5]];bb,pe:=IsConjugate(M12,ps,[ss[u][1]:u in ps]);spe:=Eltseq(pe);
for j in [u:u in [1..12]|u notin ps] do ss[j]:=[u:u in ss[j]|u ne spe[j]];end for;
end for;Append(~SS6,ss);end for;

OSS:=SS6;SS6:=[];
for uu in OSS do pp:=[i:i in [1..12]|#uu[i] eq 1];npp:=#pp;ss:=uu;
for i1 in [1..npp],i2 in [i1+1..npp],i3 in [i2+1..npp],i4 in [i3+1..npp],i5 in [i4+1..npp] do
ps:=[pp[u]:u in [i1,i2,i3,i4,i5]];bb,pe:=IsConjugate(M12,ps,[ss[u][1]:u in ps]);spe:=Eltseq(pe);
for j in [u:u in [1..12]|u notin ps] do ss[j]:=[u:u in ss[j]|u ne spe[j]];end for;
end for;Append(~SS6,ss);end for;

{{#u:u in ss}:ss in SS6};
OSS eq SS6;



"";
NSS:=[];
for ss in SS6 do if {#u:u in ss} eq {1} then Append(~NSS,ss);
elif 0 notin {#u:u in ss} then
j:=Min({i:i in [1..12]|#ss[i] ne 1});
for k in ss[j] do uu:=ss;uu[j]:=[k];Append(~NSS,uu);end for;
end if;end for;
print #NSS;
// SS7:=NSS;

cou:=0;OSS:=[];
while OSS ne NSS do
OSS:=NSS;NSS:=[];
for uu in OSS do pp:=[i:i in [1..12]|#uu[i] eq 1];npp:=#pp;ss:=uu;
for i1 in [1..npp],i2 in [i1+1..npp],i3 in [i2+1..npp],i4 in [i3+1..npp],i5 in [i4+1..npp] do
ps:=[pp[u]:u in [i1,i2,i3,i4,i5]];bb,pe:=IsConjugate(M12,ps,[ss[u][1]:u in ps]);
if bb then spe:=Eltseq(pe);
for j in [u:u in [1..12]|u notin ps] do ss[j]:=[u:u in ss[j]|u ne spe[j]];end for;end if;
end for;Append(~NSS,ss);end for;
NSS:=[ss:ss in NSS|0 notin {#u:u in ss}];
cou+:=1;cou,"\t",#OSS,"\t",#NSS;//,"\t",{{#u:u in ss}:ss in NSS};;
end while;

SS7:=Setseq(Set(NSS));#NSS,"\t",#SS7;
{{#u:u in ss}:ss in SS7};
SSC:=SSC cat [ss:ss in SS7|{#u:u in ss} eq {1}];
SS7:=[ss:ss in SS7|{#u:u in ss} ne {1}];
SSC:=Setseq(Set(SSC));
#SSC,"\t",#SS7;
// OSS eq SS7;




"";
NSS:=[];
for ss in SS7 do if {#u:u in ss} eq {1} then Append(~NSS,ss);
elif 0 notin {#u:u in ss} then
j:=Min({i:i in [1..12]|#ss[i] ne 1});
for k in ss[j] do uu:=ss;uu[j]:=[k];Append(~NSS,uu);end for;
end if;end for;
print #NSS;
// SS8:=NSS;

cou:=0;OSS:=[];
while OSS ne NSS do
OSS:=NSS;NSS:=[];
for uu in OSS do pp:=[i:i in [1..12]|#uu[i] eq 1];npp:=#pp;ss:=uu;
for i1 in [1..npp],i2 in [i1+1..npp],i3 in [i2+1..npp],i4 in [i3+1..npp],i5 in [i4+1..npp] do
ps:=[pp[u]:u in [i1,i2,i3,i4,i5]];bb,pe:=IsConjugate(M12,ps,[ss[u][1]:u in ps]);
if bb then spe:=Eltseq(pe);
for j in [u:u in [1..12]|u notin ps] do ss[j]:=[u:u in ss[j]|u ne spe[j]];end for;end if;
end for;Append(~NSS,ss);end for;
NSS:=[ss:ss in NSS|0 notin {#u:u in ss}];
cou+:=1;cou,"\t",#OSS,"\t",#NSS;//,"\t",{{#u:u in ss}:ss in NSS};;
end while;

SS8:=Setseq(Set(NSS));#NSS,"\t",#SS8;
{{#u:u in ss}:ss in SS8};
SSC:=SSC cat [ss:ss in SS8|{#u:u in ss} eq {1}];
SS8:=[ss:ss in SS8|{#u:u in ss} ne {1}];
SSC:=Setseq(Set(SSC));
#SSC,"\t",#SS8;




"";
NSS:=[];
for ss in SS8 do if {#u:u in ss} eq {1} then Append(~NSS,ss);
elif 0 notin {#u:u in ss} then
j:=Min({i:i in [1..12]|#ss[i] ne 1});
for k in ss[j] do uu:=ss;uu[j]:=[k];Append(~NSS,uu);end for;
end if;end for;
print #NSS;
// SS9:=NSS;

cou:=0;OSS:=[];
while OSS ne NSS do
OSS:=NSS;NSS:=[];
for uu in OSS do pp:=[i:i in [1..12]|#uu[i] eq 1];npp:=#pp;ss:=uu;
for i1 in [1..npp],i2 in [i1+1..npp],i3 in [i2+1..npp],i4 in [i3+1..npp],i5 in [i4+1..npp] do
ps:=[pp[u]:u in [i1,i2,i3,i4,i5]];bb,pe:=IsConjugate(M12,ps,[ss[u][1]:u in ps]);
if bb then spe:=Eltseq(pe);
for j in [u:u in [1..12]|u notin ps] do ss[j]:=[u:u in ss[j]|u ne spe[j]];end for;end if;
end for;Append(~NSS,ss);end for;
NSS:=[ss:ss in NSS|0 notin {#u:u in ss}];
cou+:=1;cou,"\t",#OSS,"\t",#NSS;//,"\t",{{#u:u in ss}:ss in NSS};;
end while;

SS9:=Setseq(Set(NSS));#NSS,"\t",#SS9;
{{#u:u in ss}:ss in SS9};
SSC:=SSC cat [ss:ss in SS9|{#u:u in ss} eq {1}];
SS9:=[ss:ss in SS9|{#u:u in ss} ne {1}];
SSC:=Setseq(Set(SSC));
#SSC,"\t",#SS9;




"";
NSS:=[];
for ss in SS9 do if {#u:u in ss} eq {1} then Append(~NSS,ss);
elif 0 notin {#u:u in ss} then
j:=Min({i:i in [1..12]|#ss[i] ne 1});
for k in ss[j] do uu:=ss;uu[j]:=[k];Append(~NSS,uu);end for;
end if;end for;
print #NSS;
// SS10:=NSS;

cou:=0;OSS:=[];
while OSS ne NSS do
OSS:=NSS;NSS:=[];
for uu in OSS do pp:=[i:i in [1..12]|#uu[i] eq 1];npp:=#pp;ss:=uu;
for i1 in [1..npp],i2 in [i1+1..npp],i3 in [i2+1..npp],i4 in [i3+1..npp],i5 in [i4+1..npp] do
ps:=[pp[u]:u in [i1,i2,i3,i4,i5]];bb,pe:=IsConjugate(M12,ps,[ss[u][1]:u in ps]);
if bb then spe:=Eltseq(pe);
for j in [u:u in [1..12]|u notin ps] do ss[j]:=[u:u in ss[j]|u ne spe[j]];end for;end if;
end for;Append(~NSS,ss);end for;
NSS:=[ss:ss in NSS|0 notin {#u:u in ss}];
cou+:=1;cou,"\t",#OSS,"\t",#NSS;//,"\t",{{#u:u in ss}:ss in NSS};;
end while;

SS10:=Setseq(Set(NSS));#NSS,"\t",#SS10;
{{#u:u in ss}:ss in SS10};
SSC:=SSC cat [ss:ss in SS10|{#u:u in ss} eq {1}];
SS10:=[ss:ss in SS10|{#u:u in ss} ne {1}];
SSC:=Setseq(Set(SSC));
#SSC,"\t",#SS10;




"";
NSS:=[];
for ss in SS10 do if {#u:u in ss} eq {1} then Append(~NSS,ss);
elif 0 notin {#u:u in ss} then
j:=Min({i:i in [1..12]|#ss[i] ne 1});
for k in ss[j] do uu:=ss;uu[j]:=[k];Append(~NSS,uu);end for;
end if;end for;
print #NSS;
// SS11:=NSS;

cou:=0;OSS:=[];
while OSS ne NSS do
OSS:=NSS;NSS:=[];
for uu in OSS do pp:=[i:i in [1..12]|#uu[i] eq 1];npp:=#pp;ss:=uu;
for i1 in [1..npp],i2 in [i1+1..npp],i3 in [i2+1..npp],i4 in [i3+1..npp],i5 in [i4+1..npp] do
ps:=[pp[u]:u in [i1,i2,i3,i4,i5]];bb,pe:=IsConjugate(M12,ps,[ss[u][1]:u in ps]);
if bb then spe:=Eltseq(pe);
for j in [u:u in [1..12]|u notin ps] do ss[j]:=[u:u in ss[j]|u ne spe[j]];end for;end if;
end for;Append(~NSS,ss);end for;
NSS:=[ss:ss in NSS|0 notin {#u:u in ss}];
cou+:=1;cou,"\t",#OSS,"\t",#NSS;//,"\t",{{#u:u in ss}:ss in NSS};;
end while;

SS11:=Setseq(Set(NSS));#NSS,"\t",#SS11;
{{#u:u in ss}:ss in SS11};
SSC:=SSC cat [ss:ss in SS11|{#u:u in ss} eq {1}];
SS11:=[ss:ss in SS11|{#u:u in ss} ne {1}];
SSC:=Setseq(Set(SSC));
#SSC,"\t",#SS11;




"";
NSS:=[];
for ss in SS11 do if {#u:u in ss} eq {1} then Append(~NSS,ss);
elif 0 notin {#u:u in ss} then
j:=Min({i:i in [1..12]|#ss[i] ne 1});
for k in ss[j] do uu:=ss;uu[j]:=[k];Append(~NSS,uu);end for;
end if;end for;
print #NSS;
// SS12:=NSS;

cou:=0;OSS:=[];
while OSS ne NSS do
OSS:=NSS;NSS:=[];
for uu in OSS do pp:=[i:i in [1..12]|#uu[i] eq 1];npp:=#pp;ss:=uu;
for i1 in [1..npp],i2 in [i1+1..npp],i3 in [i2+1..npp],i4 in [i3+1..npp],i5 in [i4+1..npp] do
ps:=[pp[u]:u in [i1,i2,i3,i4,i5]];bb,pe:=IsConjugate(M12,ps,[ss[u][1]:u in ps]);
if bb then spe:=Eltseq(pe);
for j in [u:u in [1..12]|u notin ps] do ss[j]:=[u:u in ss[j]|u ne spe[j]];end for;end if;
end for;Append(~NSS,ss);end for;
NSS:=[ss:ss in NSS|0 notin {#u:u in ss}];
cou+:=1;cou,"\t",#OSS,"\t",#NSS;//,"\t",{{#u:u in ss}:ss in NSS};;
end while;

SS12:=Setseq(Set(NSS));#NSS,"\t",#SS12;
{{#u:u in ss}:ss in SS12};
SSC:=SSC cat [ss:ss in SS12|{#u:u in ss} eq {1}];
SS12:=[ss:ss in SS12|{#u:u in ss} ne {1}];
SSC:=Setseq(Set(SSC));
#SSC,"\t",#SS12;




SS:=[[u[1]:u in ss]:ss in SSC];
Sort(~SS);
#SS,"\t",#Set(SS);
// The (interesting) yellow words are of course in SS.