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<cstring>
#include<algorithm>
#include<cmath>
#define SF scanf
#define PF printf
#define MAXS 2097162
#define MAXM 30
#define MOD 1000000007
using namespace std;
typedef long long ll;
int dp[2][MAXM][MAXS],ans;
int nxt[MAXS][MAXM],mp[MAXM][MAXM];
int n,m,p;
int main(){
	SF("%d%d%d",&n,&m,&p);
	n--;
	for(int i=0;i<(1<<n);i++){
		int t=-1;
		for(int j=n-1;j>=0;j--){
			if((i&(1<<j)))
				t=j;
			nxt[i][j]=t;
		}
	}
	for(int i=1;i<=m;i++)
		for(int j=0;j<n;j++)
			mp[i][j]=-1;
	int x,y,z;
	for(int i=1;i<=p;i++){
		SF("%d%d%d",&x,&y,&z);
		mp[x][y-1]=z;	
	}
	dp[0][n][0]=1;
	int now=1;
	for(int i=1;i<=m;i++){
		for(int mask=0;mask<(1<<n);mask++)
			dp[now][0][mask]=dp[now^1][n][mask];
		for(int j=0;j<=n;j++)
			for(int mask=0;mask<(1<<n);mask++)
				dp[now^1][j][mask]=0;
		for(int j=0;j<n;j++)
			for(int mask=0;mask<(1<<n);mask++)
				if(dp[now][j][mask]){
					for(int k=0;k<=1;k++){
						if(mp[i][j]!=-1&&mp[i][j]!=k)
							continue;
						int t=min((mask&(1<<j)),1);
						if(k==0&&t==1)
							continue;
						if(k==t)
							dp[now][j+1][mask]=(dp[now][j+1][mask]+dp[now][j][mask])%MOD;
						else{
							int r=nxt[mask][j];
							if(r==-1)
								dp[now][j+1][mask+(1<<j)]=(dp[now][j+1][mask+(1<<j)]+dp[now][j][mask])%MOD;
							else
								dp[now][j+1][mask+(1<<j)-(1<<r)]=(dp[now][j+1][mask+(1<<j)-(1<<r)]+dp[now][j][mask])%MOD;
						}
					}
				}
		now^=1;	
	}
	for(int mask=0;mask<(1<<n);mask++)
		ans=(ans+dp[now^1][n][mask])%MOD;
	PF("%d",ans);
}