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>
using namespace std;
const int N=6e5+10,p=1e9+7;
void update(int &x,int y){x+=y;if (x>=p) x-=p;}
int n,m,k,U,must[30],state[30],lowbit[N];
int dp[30][N];
void trans(int *x,int *y){
    static int dp[30][N];
    for (int i=0;i<n;i++)
    for (int S=0;S<U;S++)
        dp[i][S]=0;
    for (int S=0;S<U;S++) dp[0][S]=x[S];
    for (int i=0;i<n-1;i++)
    for (int S=0;S<U;S++){
        int v=dp[i][S];
        update(dp[i+1][S],v);
        if (!(S>>i&1)){
            int T=S;
            T^=lowbit[T>>i]<<i;
            T^=1<<i;
            T&=(U-1);
            update(dp[i+1][T],v);
        }
    }
    for (int S=0;S<U;S++) y[S]=dp[n-1][S];
}
int main()
{
	scanf("%d%d%d",&n,&m,&k);
	for (int i=1;i<=k;i++){
		int a,b,c;
		scanf("%d%d%d",&a,&b,&c);b--;
		must[a]|=(1<<b);
		state[a]|=(c<<b);
	}
	U=1<<(n-1);
	for (int S=0;S<U;S++) lowbit[S]=S&-S;
	dp[0][0]=1;
	for (int i=1;i<=m;i++){
        trans(dp[i-1],dp[i]);
        for (int S=0;S<U;S++)
        if (must[i]&S^state[i]) dp[i][S]=0;
	}
	int ans=0;
	for (int S=0;S<U;S++) update(ans,dp[m][S]);
	printf("%d\n",ans);
	return 0;
}