"""Floyd-Warshall all-pairs shortest paths."""
from __future__ import annotations


def floyd_warshall(n, edges):
    INF = float("inf")
    dist = [[INF] * n for _ in range(n)]
    nxt = [[None] * n for _ in range(n)]
    for i in range(n):
        dist[i][i] = 0
        nxt[i][i] = i
    for (u, v, w) in edges:
        if w < dist[u][v]:
            dist[u][v] = w
            nxt[u][v] = v
    for k in range(n):
        dk = dist[k]
        nk = nxt[k]
        for i in range(n):
            dik = dist[i][k]
            if dik == INF:
                continue
            di = dist[i]
            ni = nxt[i]
            for j in range(n):
                if dk[j] == INF:
                    continue
                nd = dik + dk[j]
                if nd < di[j]:
                    di[j] = nd
                    ni[j] = ni[k]
    for i in range(n):
        if dist[i][i] < 0:
            raise ValueError("negative cycle detected")
    return dist, nxt


def reconstruct_path(nxt, s, t):
    if nxt[s][t] is None:
        return None
    path = [s]
    cur = s
    seen = set()
    while cur != t:
        if cur in seen:
            return None
        seen.add(cur)
        cur = nxt[cur][t]
        if cur is None:
            return None
        path.append(cur)
        if len(path) > len(nxt) + 1:
            return None
    return path