CF1366F.Jog Around The Graph

You are given a simple weighted connected undirected graph, consisting of $n$ vertices and $m$ edges.

A path in the graph of length $k$ is a sequence of $k+1$ vertices $v_1, v_2, \dots, v_{k+1}$ such that for each $i$ $(1 \le i \le k)$ the edge $(v_i, v_{i+1})$ is present in the graph. A path from some vertex $v$ also has vertex $v_1=v$ . Note that edges and vertices are allowed to be included in the path multiple times.

The weight of the path is the total weight of edges in it.

For each $i$ from $1$ to $q$ consider a path from vertex $1$ of length $i$ of the maximum weight. What is the sum of weights of these $q$ paths?

Answer can be quite large, so print it modulo $10^9+7$ .

The first line contains a three integers $n$ , $m$ , $q$ ( $2 \le n \le 2000$ ; $n - 1 \le m \le 2000$ ; $m \le q \le 10^9$ ) — the number of vertices in the graph, the number of edges in the graph and the number of lengths that should be included in the answer.

Each of the next $m$ lines contains a description of an edge: three integers $v$ , $u$ , $w$ ( $1 \le v, u \le n$ ; $1 \le w \le 10^6$ ) — two vertices $v$ and $u$ are connected by an undirected edge with weight $w$ . The graph contains no loops and no multiple edges. It is guaranteed that the given edges form a connected graph.

Print a single integer — the sum of the weights of the paths from vertex $1$ of maximum weights of lengths $1, 2, \dots, q$ modulo $10^9+7$ .

Here is the graph for the first example:

Some maximum weight paths are:

• length $1$ : edges $(1, 7)$ — weight $3$ ;
• length $2$ : edges $(1, 2), (2, 3)$ — weight $1+10=11$ ;
• length $3$ : edges $(1, 5), (5, 6), (6, 4)$ — weight $2+7+15=24$ ;
• length $4$ : edges $(1, 5), (5, 6), (6, 4), (6, 4)$ — weight $2+7+15+15=39$ ;
• $\dots$

So the answer is the sum of $25$ terms: $3+11+24+39+\dots$

In the second example the maximum weight paths have weights $4$ , $8$ , $12$ , $16$ and $20$ .