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;

#define FOR(i,a,b) for(int i=int(a);i<int(b);i++)
#define REP(i,b) FOR(i,0,b)

int read(){
	int i;
	scanf("%d",&i);
	return i;
}

const int mod=1000000007;
void add(int&a,int b){
	a+=b;
	if(a>=mod)a-=mod;
}

const int S=19;
const int M=20;
int dp[1<<S];
int rest_pos[M],rest_val[M];

int main(){
	int n=read()-1,m=read(),k=read();
	REP(i,k){
		int a=read()-1,b=n-read(),c=read();
		rest_pos[a]+=(1<<b);
		rest_val[a]+=(c<<b);
	}
	REP(bit,1<<n)
		if((bit&rest_pos[0])==rest_val[0])
			dp[bit]=1;
	REP(i,m){
		if(i>0){
			REP(bit,(1<<n)-1)
				add(dp[bit|(bit+1)],dp[bit]);
			for(int j=0;j<n-1;j++){
				int w=1<<j;
				REP(bit,(1<<n)-w)if(bit&w)
					add(dp[(bit^w)|(bit+w)],dp[bit]);
			}
		}
		REP(bit,1<<n)
			if((bit&rest_pos[i])!=rest_val[i])
				dp[bit]=0;
	}
	int ans=0;
	REP(bit,1<<n)
		add(ans,dp[bit]);
	cout<<ans<<endl;
}