Vibrating Paths (Challenge)

You are given an undirected weighted graph with $N$ vertices (numbered $1$ through $N$) and $M$ edges (numbered $1$ through $M$). For each valid $i$, the $i$-th edge has weight $S_i$ and connects vertices $u_i$ and $v_i$. The weights of all edges are pairwise distinct. Let's denote an edge that connects vertices $u$ and $v$ by $(u,v)$. A *vibrating path* is a sequence of distinct vertices $V_1, V