:- op(700, xfx, l).
:- op(700, xfx, r).
From l To :- From to To-_.
From r To :- From to _-To.

answer(Answer) :-
    findall(Node, (Node to _, atom_chars(Node, [_, _, a])), Starts),
    routes(Starts, [Route1 | RestOfRoutes]),
    foldl(unify2, RestOfRoutes, Route1, _-(_-Answer-_)).

% unify2 combines 2 routes into one with its own stride-offsets-dests
unify2(Route1, Route2, NewRoute) :-
    writef('Combining %t + %t', [Route1, Route2]),
    once(findnsols(2, S, converge(Route1, Route2, S), SolutionPair)),
    SolutionPair = [NewA-LenA-NewZ, _-LenB-_],
    NewStride is LenB - LenA,
    NewRoute = NewA-(NewStride-LenA-[0-NewZ]),
    writef(' --> %t\n', [NewRoute]).

% Len = Stride1*X1 + Offset1 + Dest1 = Stride2*X2 + Offset2 + Dest2
% For performance, Route1's Stride should =< Route2's Stride
converge(Route1, Route2, NewA-Len-NewZ) :-
    Route1 = A1-(Stride1-Offset1-Dests1),
    Route2 = A2-(Stride2-Offset2-Dests2),
    natnum(X2),
    pick([Dests1, Dests2], [Dest1-Z1, Dest2-Z2]),
    0 is (Stride2*X2 + Offset2 + Dest2 - Offset1 - Dest1) mod Stride1,
    Len is Stride2*X2 + Offset2 + Dest2,
    atom_concat(A1, A2, NewA), atom_concat(Z1, Z2, NewZ).

routes(Starts, Routes) :-
    maplist([S, S-Route]>>(route_at(S, Route)), Starts, Routes).

% route_at(N, Node, Dests, AllDests) means going N steps will arrive at Node
% passing through Dests.
route_at(Start, Route) :- route_at(0, Start, [], Route).
route_at(N, FirstZ, [FirstZN-FirstZ | Zs], Stride-Offset-AllZs) :-
    Stride is N - FirstZN, direction_looped(Stride),
    Offset is FirstZN,
    maplist({Offset}/[N-X, M-X]>>(M is N-Offset), [FirstZN-FirstZ | Zs], AllZs),
    !.
route_at(N, Node, Dests, AllDests) :-
    (   atom_chars(Node, [_, _, z])
    ->  append(Dests, [N-Node], NextDests)
    ;   NextDests = Dests),
    Nplus1 is N + 1, step_at(N, Step), call(Step, Node, NextNode),
    route_at(Nplus1, NextNode, NextDests, AllDests).

% Step is the N-th step (counting starts from 0).
step_at(N, Step) :-
    direction_list(Dir), length(Dir, DirLen), Remainder is N mod DirLen,
    nth0(Remainder, Dir, Step).

direction_looped(Len) :- direction_list(D), length(D, DLen), 0 is Len mod DLen.
direction_list(Dir) :- direction(Str), atom_chars(Str, Dir).

% pick one item from each sublist of ListOfLists & put them into Items in order.
% [[1,2,3], [4], [5,6]] -> [1,4,5]; [1,4,6]; [2,4,5]; [2,4,6]; [3,4,5]; [3,4,6].
pick(ListOfLists, Items) :-
    maplist([SubList, X]>>(member(X, SubList)), ListOfLists, Items).

natnum(0).
natnum(N) :- natnum(N0), N is N0 + 1.