The Classic Problem

Description:

Input:

The first line of the input contains two space-separated integers — n and m (1 ≤ n ≤ 10⁵; 0 ≤ m ≤ 10⁵).

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 ≤ 10⁵). 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.

Output:

In the first line print the remainder after dividing the length of the shortest path by 1000000007 (10⁹ + 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.

Sample Input:

4 4
1 4 2
1 2 0
2 3 0
3 4 0
1 4

Sample Output:

3
4
1 2 3 4 

Sample Input:

4 3
1 2 4
2 3 5
3 4 6
1 4

Sample Output:

112
4
1 2 3 4 

Sample Input:

4 2
1 2 0
3 4 1
1 4

Sample Output:

-1

Note:

A path from vertex s to vertex t is a sequence v₀, ..., v_k, such that v₀ = 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.