# OK to post

# PART 2: qSR() PROCEDURE:
# -----------------------

qSR:=proc(q,INI,L,n) local r,i,M,ng:
	r:=nops(L);

	if nops(INI)<>L[-1][2] then
		return FAIL;
	fi;

	if not (sort(INI)[-1] < q and sort(INI)[1] >= 0) then
		return FAIL;
	fi;

	if not (type(L, list) and {seq(type(L[i], list), i=1..nops(L))} = {true} 
	                      and {seq(type(L[i][1], posint), i=1..nops(L))} = {true} 
						  and {seq(type(L[i][2], posint), i=1..nops(L))} = {true} 
						  and sort([seq(L[i][1], i=1..nops(L))])[-1] < q 
						  and sort([seq(L[i][1], i=1..nops(L))])[1] >= 0 
						  and sort([seq(L[i][2], i=1..nops(L))]) = [seq(L[i][2], i=1..nops(L))]) then
		return FAIL;
	fi;

	if not type(n,posint) then
		return FAIL;
	fi;

	M:=INI;

	while(nops(M))<n do
		ng:=add(L[i][1]*M[-L[i][2]],i=1..r) mod q;
		M:=[op(M),ng];
	od;

	return M;
end:

IsPer:=proc(L,t) local i:
	if {seq(L[i]-L[i+t],i=1..nops(L)-t)}={0} then
		true;
	else
		false;
	fi;
end;

FindPer:=proc(L) local t:
	for t from 1 to trunc(nops(L)/2) while not IsPer(L,t) do od;

	if t=trunc(nops(L)/2)+1 then
		return FAIL;
	else
		return t;
	fi;
end;

# PART 2: PERIODS:
# -----------------------

FindPer(qSR(3,[1,2,1],[[1,2],[2,3]],10000));
FindPer(qSR(5,[1,2,4,1,2],[[1,2],[2,5]],10000));
FindPer(qSR(11,[3,1,2,4,3],[[1,2],[2,5]],10000));
FindPer(qSR(5,[1,0,0,0,0,0],[[3,4],[2,5],[3,6]],10000));