B. Boboniu Walks on Graph
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output

Boboniu has a directed graph with $$$n$$$ vertices and $$$m$$$ edges.

The out-degree of each vertex is at most $$$k$$$.

Each edge has an integer weight between $$$1$$$ and $$$m$$$. No two edges have equal weights.

Boboniu likes to walk on the graph with some specific rules, which is represented by a tuple $$$(c_1,c_2,\ldots,c_k)$$$. If he now stands on a vertex $$$u$$$ with out-degree $$$i$$$, then he will go to the next vertex by the edge with the $$$c_i$$$-th $$$(1\le c_i\le i)$$$ smallest weight among all edges outgoing from $$$u$$$.

Now Boboniu asks you to calculate the number of tuples $$$(c_1,c_2,\ldots,c_k)$$$ such that

  • $$$1\le c_i\le i$$$ for all $$$i$$$ ($$$1\le i\le k$$$).
  • Starting from any vertex $$$u$$$, it is possible to go back to $$$u$$$ in finite time by walking on the graph under the described rules.
Input

The first line contains three integers $$$n$$$, $$$m$$$ and $$$k$$$ ($$$2\le n\le 2\cdot 10^5$$$, $$$2\le m\le \min(2\cdot 10^5,n(n-1) )$$$, $$$1\le k\le 9$$$).

Each of the next $$$m$$$ lines contains three integers $$$u$$$, $$$v$$$ and $$$w$$$ $$$(1\le u,v\le n,u\ne v,1\le w\le m)$$$, denoting an edge from $$$u$$$ to $$$v$$$ with weight $$$w$$$. It is guaranteed that there are no self-loops or multiple edges and each vertex has at least one edge starting from itself.

It is guaranteed that the out-degree of each vertex is at most $$$k$$$ and no two edges have equal weight.

Output

Print one integer: the number of tuples.

Examples
Input
4 6 3
4 2 1
1 2 2
2 4 3
4 1 4
4 3 5
3 1 6
Output
2
Input
5 5 1
1 4 1
5 1 2
2 5 3
4 3 4
3 2 5
Output
1
Input
6 13 4
3 5 1
2 5 2
6 3 3
1 4 4
2 6 5
5 3 6
4 1 7
4 3 8
5 2 9
4 2 10
2 1 11
6 1 12
4 6 13
Output
1
Note

For the first example, there are two tuples: $$$(1,1,3)$$$ and $$$(1,2,3)$$$. The blue edges in the picture denote the $$$c_i$$$-th smallest edges for each vertex, which Boboniu chooses to go through.

For the third example, there's only one tuple: $$$(1,2,2,2)$$$.

The out-degree of vertex $$$u$$$ means the number of edges outgoing from $$$u$$$.