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Contest: Task: Related: TaskG TaskI

Score : $600$ points

Problem Statement

Given is an undirected graph with $N$ vertices and $M$ edges. The $i$-th edge connects Vertices $A_i$ and $B_i$ and has a weight of $C_i$.

Initially, all vertices are painted black. You can do the following operation at most $K$ times.

  • Operation: choose any vertex $v$ and any integer $x$. Paint red all vertices reachable from the vertex $v$ by traversing edges whose weights are at most $x$, including $v$ itself.

How many sets can be the set of vertices painted red after the operations?
Find the count modulo $998244353$.

Constraints

  • $2 \leq N \leq 10^5$
  • $0 \leq M \leq 10^5$
  • $1 \leq K \leq 500$
  • $1 \leq A_i,B_i \leq N$
  • $1 \leq C_i \leq 10^9$
  • All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$
$A_1$ $B_1$ $C_1$
$\vdots$
$A_M$ $B_M$ $C_M$

Output

Print the answer.

Sample Input 1

3 2 1
1 2 1
2 3 2

Sample Output 1

6

For example, an operation with $(v,x)=(2,1)$ paints Vertices $1,2$ red, and an operation with $(v,x)=(1,0)$ paints Vertex $1$.

After at most one operation, the set of vertices painted red can be one of the following six: $\{\},\{1\},\{2\},\{3\},\{1,2\},\{1,2,3\}$.

Sample Input 2

5 0 2

Sample Output 2

16

The given graph may not be connected.

Sample Input 3

6 8 2
1 2 1
2 3 2
3 4 3
4 5 1
5 6 2
6 1 3
1 2 10
1 1 100

Sample Output 3

40

The given graph may have multi-edges and self-loops.