Title
Problem Statement
We have a graph with N vertices and M edges, and there are two people on the graph: Takahashi and Aoki.
The i-th edge connects Vertex Ui and Vertex Vi.
The time it takes to traverse this edge is Di minutes, regardless of direction and who traverses the edge (Takahashi or Aoki).
Takahashi departs Vertex S and Aoki departs Vertex T at the same time. Takahashi travels to Vertex T and Aoki travels to Vertex S, both in the shortest time possible.
Find the number of the pairs of ways for Takahashi and Aoki to choose their shortest paths such that they never meet (at a vertex or on an edge) during the travel, modulo 109 + 7.
Constraints
- 1 ≤ N ≤ 100 000
- 1 ≤ M ≤ 200 000
- 1 ≤ S, T ≤ N
- S ≠ T
- 1 ≤ Ui, Vi ≤ N (1 ≤ i ≤ M)
- 1 ≤ Di ≤ 109 (1 ≤ i ≤ M)
- If i ≠ j, then (Ui, Vi) ≠ (Uj, Vj) and (Ui, Vi) ≠ (Vj, Uj).
- Ui ≠ Vi (1 ≤ i ≤ M)
- Di are integers.
- The given graph is connected.
Input
Input is given from Standard Input in the following format:
N M
S T
U1 V1 D1
U2 V2 D2
:
UM VM DM
Output
Print the answer.
Sample Input 1
4 4
1 3
1 2 1
2 3 1
3 4 1
4 1 1
Sample Output 1
2
There are two ways to choose shortest paths that satisfies the condition:
- Takahashi chooses the path 1 \rightarrow 2 \rightarrow 3, and Aoki chooses the path 3 \rightarrow 4 \rightarrow 1.
- Takahashi chooses the path 1 \rightarrow 4 \rightarrow 3, and Aoki chooses the path 3 \rightarrow 2 \rightarrow 1.
Sample Input 2
3 3
1 3
1 2 1
2 3 1
3 1 2
Sample Output 2
2
Sample Input 3
3 3
1 3
1 2 1
2 3 1
3 1 2
Sample Output 3
2
Sample Input 4
8 13
4 2
7 3 9
6 2 3
1 6 4
7 6 9
3 8 9
1 2 2
2 8 12
8 6 9
2 5 5
4 2 18
5 3 7
5 1 515371567
4 8 6
Sample Output 4
6