:- table direction_loop/1.

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

answer(Answer) :-
    starts(Starts),
    routes(Starts, Routes),
    collapse_routes(Routes, _-(Answer-_-_)).

collapse_routes([X], X).
collapse_routes(Routes, Collapsed) :-
    sort(2, @=<, Routes, [Route1, Route2 | CdrRoutes]),
    unify2(Route1, Route2, 0-0-0-0, [], NewRoute),
    collapse_routes([NewRoute | CdrRoutes], Collapsed).

unify2(Route1, Route2, N1-Term1-N2-Term2, Founds, NewRoute) :-
    Route1 = A1-(Stride1-Offset1-Loop1),
    Route2 = A2-(Stride2-Offset2-Loop2),
    nth0(Term1, Loop1, C1-_),
    nth0(Term2, Loop2, C2-_),
    Z1 is Offset1 + Stride1*N1 + C1,
    Z2 is Offset2 + Stride2*N2 + C2,
    next(Route1, N1-Term1, NextN1-NextTerm1),
    (   Z1 =:= Z2
    ->  (   Founds = [OldZ]
        ->  atom_concat(A1, A2, NewA),
            NewStride is Z1 - OldZ,
            NewRoute = NewA-(NewStride-OldZ-[0-NewA]),
            !
        ;   unify2(Route1, Route2, NextN1-NextTerm1-N2-Term2, [Z1], NewRoute)
        )
    ;   Z1 < Z2
    ->  unify2(Route1, Route2, NextN1-NextTerm1-N2-Term2, Founds, NewRoute)
    ;   next(Route2, N2-Term2, NextN2-NextTerm2),
        unify2(Route1, Route2, N1-Term1-NextN2-NextTerm2, Founds, NewRoute)
    ).

next(_-(_-_-Loop), NLoops-Term, NextNLoops-NextTerm) :-
    length(Loop, LoopLen),
    (   Term < LoopLen - 1
    ->  NextNLoops is NLoops, NextTerm is Term + 1
    ;   NextNLoops is NLoops + 1, NextTerm is 0).

routes(Starts, Routes) :-
    maplist([S, S-Route]>>(zloop([], S, 0, [], Route)), Starts, Routes).

zloop(_, _, _, Zs, Stride-Offset-Loop) :-
    Zs = [FirstZIndex-Z | _], reverse(Zs, [LastZIndex-Z | ReversedZs]),
    DeltaZ is LastZIndex - FirstZIndex, DeltaZ =\= 0,
    direction_len(Len),
    divmod(DeltaZ, Len, _, 0),
    Offset = FirstZIndex, Stride = DeltaZ,
    foldl([Idx-Z, NewIdx-Z, Off, Off]>>(NewIdx is Idx - Off),
          ReversedZs, ReversedLoop, Offset, _),
    reverse(ReversedLoop, Loop),
    !.

zloop(Directions, Node, Index, Zs, Route) :-
    (   is_end(Node)
    ->  append(Zs, [Index-Node], NewZs)
    ;   NewZs = Zs
    ),
    next_step(Directions, Move, Remain),
    G =.. [Move, Node, To], G,
    NewIndex is Index + 1,
    zloop(Remain, To, NewIndex, NewZs, Route).

starts(Starts) :- findall(X, X to _, Nodes), include(is_start, Nodes, Starts).

is_start(Node) :- atom_chars(Node, [_, _, a]).
is_end(Node) :- atom_chars(Node, [_, _, z]).

next_step([Move | Remain], Move, Remain).
next_step([], Move, Remain) :- direction_list([Move | Remain]).

direction_list(Dir) :- direction(Str), atom_chars(Str, Dir).
direction_len(Len) :- direction_list(D), length(D, Len).