A9624. The Classic Problem

You are given a weighted undirected graph on $n$ vertices and $m$ edges. Find the shortest path from vertex $s$ to vertex $t$ or else state that such path doesn't exist.

The first line of the input contains two space-separated integers — $n$ and $m$ ( $1<=n<=10^{5}$ ; $0<=m<=10^{5}$ ).

Next $m$ lines contain the description of the graph edges. The $i$ -th line contains three space-separated integers — $u_{i}$ , $v_{i}$ , $x_{i}$ ( $1<=u_{i},v_{i}<=n$ ; $0<=x_{i}<=10^{5}$ ). That means that vertices with numbers $u_{i}$ and $v_{i}$ are connected by edge of length $2^{x_{i}}$ (2 to the power of $x_{i}$ ).

The last line contains two space-separated integers — the numbers of vertices $s$ and $t$ .

The vertices are numbered from $1$ to $n$ . The graph contains no multiple edges and self-loops.

In the first line print the remainder after dividing the length of the shortest path by $1000000007 (10^{9}+7)$ if the path exists, and -1 if the path doesn't exist.

If the path exists print in the second line integer $k$ — the number of vertices in the shortest path from vertex $s$ to vertex $t$ ; in the third line print $k$ space-separated integers — the vertices of the shortest path in the visiting order. The first vertex should be vertex $s$ , the last vertex should be vertex $t$ . If there are multiple shortest paths, print any of them.

A path from vertex $s$ to vertex $t$ is a sequence $v_{0}$ , ..., $v_{k}$ , such that $v_{0}=s$ , $v_{k}=t$ , and for any $i$ from 0 to $k-1$ vertices $v_{i}$ and $v_{i+1}$ are connected by an edge.

The length of the path is the sum of weights of edges between $v_{i}$ and $v_{i+1}$ for all $i$ from 0 to $k-1$ .

The shortest path from $s$ to $t$ is the path which length is minimum among all possible paths from $s$ to $t$ .