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<bits/stdc++.h>
#define maxn 1124291
#define mod 1000000007
using namespace std;
int f[23][maxn],g[23][23],h[maxn][23];
int main(){
  int n,m,k,x,y,z;
  scanf("%d%d%d",&n,&m,&k);
  memset(g,-1,sizeof(g));
  for(int i=1;i<=k;i++)
    scanf("%d%d%d",&x,&y,&z),g[x][y]=z;
  for(int i=0;i<1<<n;i++){
    for(int j=0;j<n-1;j++)
      if(!(i>>j&1)){
	for(k=j+1;k<n-1;k++)
	  if(i>>k&1){h[i][j]=i^(1<<j)^(1<<k);break;}
	if(!h[i][j]) h[i][j]=i^(1<<j);
      }
  } f[n-1][0]=1;
  for(int i=1;i<=m;i++){
    for(int j=0;j<1<<n;j++) f[0][j]=f[n-1][j],f[n-1][j]=0;
    for(int j=0;j<n-1;j++)
      for(k=0;k<1<<n;k++)
	if(f[j][k]){
	  if(g[i][j+1]!=((k>>j&1)^1)) f[j+1][k]=(f[j+1][k]+f[j][k])%mod;
	  if(g[i][j+1] && !(k>>j&1)) f[j+1][h[k][j]]=(f[j+1][h[k][j]]+f[j][k])%mod;
	  f[j][k]=0;
	}
  }
  int ans=0;
  for(int i=0;i<1<<n;i++) ans=(ans+f[n-1][i])%mod;
  printf("%d",ans);
  return 0;
}