Problem:
Score : 1600 points

Problem Statement

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).

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.

Notes

Before submission, it is strongly recommended to measure the execution time of your code using "Custom Test".

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.

Sample Input 1

3 2 1
1 2 0

Sample Output 1

6

There are six ways to draw lines, as shown below. Here, red lines represent L_1, and green lines represent L_2.

Sample Input 2

3 2 2
1 1 1
2 1 0

Sample Output 2

0

Sample Input 3

5 4 2
1 3 1
4 2 0

Sample Output 3

172

Sample Input 4

20 20 0

Sample Output 4

881396682

Solution in C++14 (GCC 5.4.1) (judge verdict: Memory Limit Exceeded)

#include<cstdio>
#include<cstring>
const int mod=1000000007;
inline void add(int &a,int b){if((a+=b)>=mod)a-=mod;}
int p[30],q[30],go[1<<21][30],put[1<<21],f[1<<21],n,m,k;
int main()
{
	scanf("%d%d%d",&n,&m,&k);
	while(k--)
	{
		int a,b,c;
		scanf("%d%d%d",&a,&b,&c);
		a--;b--;
		p[a]|=1<<b;
		q[a]|=c<<b;
	}
	n--;
	memset(go,-1,sizeof go);
	for(int t=0;t<(1<<n);t++)
	{
		int id=0;
		for(int i=0;i<n;i++)
			if(t&(1<<i))
			{
				if(i&&!(t&(1<<(i-1))))go[t][id]=t^(1<<i)^(1<<(i-1));
				id++;
			}
	}
	memset(put,-1,sizeof put);
	for(int t=0;t<(1<<n);t++)
		for(int i=n-1;i>=0;i--)
			if(!(t&(1<<i))){put[t]=t^(1<<i);break;}
	f[0]=1;
	for(int i=0;i<m;i++)
	{
		for(int t=0;t<(1<<n)-1;t++)add(f[put[t]],f[t]);
		for(int j=0;j<n;j++)
			for(int t=(1<<n)-1;t>=0;t--)
				if(go[t][j]!=-1)add(f[go[t][j]],f[t]);
		for(int t=0;t<(1<<n);t++)
			if((t&p[i])!=q[i])f[t]=0;
	}
	int ans=0;
	for(int t=0;t<(1<<n);t++)add(ans,f[t]);
	printf("%d\n",ans);
	return 0;
}