Problem: There are N(N+1)/2 dots arranged to form an equilateral triangle whose sides consist of N dots, as shown below. The j-th dot from the left in the i-th row from the top is denoted by (i, j) (1 \leq i \leq N, 1 \leq j \leq i). Also, we will call (i+1, j) immediately lower-left to (i, j), and (i+1, j+1) immediately lower-right to (i, j).

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Takahashi is drawing M polygonal lines L_1, L_2, ..., L_M by connecting these dots. Each L_i starts at (1, 1), and visits the dot that is immediately lower-left or lower-right to the current dots N-1 times. More formally, there exist X_{i,1}, ..., X_{i,N} such that:

* L_i connects the N points (1, X_{i,1}), (2, X_{i,2}), ..., (N, X_{i,N}), in this order.
* For each j=1, 2, ..., N-1, either X_{i,j+1} = X_{i,j} or X_{i,j+1} = X_{i,j}+1 holds.



Takahashi would like to draw these lines so that no part of L_{i+1} is to the left of L_{i}. That is, for each j=1, 2, ..., N, X_{1,j} \leq X_{2,j} \leq ... \leq X_{M,j} must hold.

Additionally, there are K conditions on the shape of the lines that must be followed. The i-th condition is denoted by (A_i, B_i, C_i), which means:

* If C_i=0, L_{A_i} must visit the immediately lower-left dot for the B_i-th move.
* If C_i=1, L_{A_i} must visit the immediately lower-right dot for the B_i-th move.



That is, X_{A_i, {B_i}+1} = X_{A_i, B_i} + C_i must hold.

In how many ways can Takahashi draw M polygonal lines? Find the count modulo 1000000007.

Constraints

* 1 \leq N \leq 20
* 1 \leq M \leq 20
* 0 \leq K \leq (N-1)M
* 1 \leq A_i \leq M
* 1 \leq B_i \leq N-1
* C_i = 0 or 1
* No pair appears more than once as (A_i, B_i).

Input

Input is given from Standard Input in the following format:


N M K
A_1 B_1 C_1
A_2 B_2 C_2
:
A_K B_K C_K


Output

Print the number of ways for Takahashi to draw M polygonal lines, modulo 1000000007.

Examples

Input

3 2 1
1 2 0


Output

6


Input

3 2 2
1 1 1
2 1 0


Output

0


Input

5 4 2
1 3 1
4 2 0


Output

172


Input

20 20 0


Output

881396682

Solution (in CPP):

Solution:
#include <cstdio>
#include <vector>
#include <algorithm>
const int MOD = 1000000007;
int N, M, K, f[524288], g[524288];
inline void add(int &x, int y)
{
	x = x + y < MOD ? x + y : x + y - MOD;
}
void Sanho(int *f, int d)
{
	if (d == 0)
		return;
	add(f[1 << d - 1], f[0]);
	for (int i = 1; i < 1 << d - 1; i <<= 1)
		for (int j = 0; j < i; j++)
			add(f[(1 << d - 1) + j], f[i + j]);
	Sanho(f, d - 1);
	Sanho(f + (1 << d - 1), d - 1);
}
std::vector < std::pair < int, int > > lim[20];
int main()
{
	scanf("%d%d%d", &N, &M, &K);
	N--;
	while (K--)
	{
		int a, b, c;
		scanf("%d%d%d", &a, &b, &c);
		lim[a - 1].push_back({ N - b, c });
	}
	f[0] = 1;
	for (int i = 0; i < M; i++)
	{
		Sanho(f, N);
		for (int j = 0; j < 1 << N; j++)
			for (auto k : lim[i])
				if ((j >> k.first & 1) != k.second)
					f[j] = 0;
	}
	int O = 0;
	for (int i = 0; i < 1 << N; i++)
		add(O, f[i]);
	printf("%d\n", O);
	return 0;
}