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 <algorithm>
#include <cstring>
using namespace std;

const int MOD = 1000000007;

int f[1 << 20], g[1 << 20], N, M, K;
int mat[25][25];

inline void update(int &x, int y) {
  x = (x + y) % MOD;
}

int main() {
  scanf("%d%d%d", &N, &M, &K);
  memset(mat, -1, sizeof mat);
  for (int i = 1; i <= K; ++i) {
    int A, B, C;
    scanf("%d%d%d", &A, &B, &C);
    mat[A][B] = C;
  }

  f[0] = 1;
  for (int i = 1; i <= M; ++i)
    for (int j = 1; j < N; ++j) {
      memset(g, 0, sizeof g);
      for (int mask = 0; mask < 1 << N; ++mask) {
	if (!f[mask]) continue;
	if (mat[i][j] != 1 && !(mask & (1 << j - 1))) {
	  update(g[mask], f[mask]);
	}
	if (mat[i][j] != 0) {
	  int temp = mask & ((1 << j - 1) - 1);
	  int nxt_mask = mask ^ temp;

	  if (mask & (1 << j - 1)) nxt_mask = mask;
	  else nxt_mask ^= (nxt_mask & -nxt_mask) ^ temp ^ (1 << j - 1);

	  update(g[nxt_mask], f[mask]);
	}
      }
      memcpy(f, g, sizeof f);
    }

  int res = 0;
  for (int mask = 0; mask < 1 << N; ++mask)
    update(res, f[mask]);

  printf("%d\n", res);
  return 0;
}