Problem:
Score : 300 points

Problem Statement

We have N switches with "on" and "off" state, and M bulbs. The switches are numbered 1 to N, and the bulbs are numbered 1 to M.

Bulb i is connected to k_i switches: Switch s_{i1}, s_{i2}, ..., and s_{ik_i}. It is lighted when the number of switches that are "on" among these switches is congruent to p_i modulo 2.

How many combinations of "on" and "off" states of the switches light all the bulbs?

Constraints

1 \leq N, M \leq 10

1 \leq k_i \leq N

1 \leq s_{ij} \leq N

s_{ia} \neq s_{ib} (a \neq b)

p_i is 0 or 1.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M
k_1 s_{11} s_{12} ... s_{1k_1}
:
k_M s_{M1} s_{M2} ... s_{Mk_M}
p_1 p_2 ... p_M

Output

Print the number of combinations of "on" and "off" states of the switches that light all the bulbs.

Sample Input 1

2 2
2 1 2
1 2
0 1

Sample Output 1

1

Bulb 1 is lighted when there is an even number of switches that are "on" among the following: Switch 1 and 2.

Bulb 2 is lighted when there is an odd number of switches that are "on" among the following: Switch 2.

There are four possible combinations of states of (Switch 1, Switch 2): (on, on), (on, off), (off, on) and (off, off). Among them, only (on, on) lights all the bulbs, so we should print 1.

Sample Input 2

2 3
2 1 2
1 1
1 2
0 0 1

Sample Output 2

0

Bulb 1 is lighted when there is an even number of switches that are "on" among the following: Switch 1 and 2.

Bulb 2 is lighted when there is an even number of switches that are "on" among the following: Switch 1.

Bulb 3 is lighted when there is an odd number of switches that are "on" among the following: Switch 2.

Switch 1 has to be "off" to light Bulb 2 and Switch 2 has to be "on" to light Bulb 3, but then Bulb 1 will not be lighted. Thus, there are no combinations of states of the switches that light all the bulbs, so we should print 0.

Sample Input 3

5 2
3 1 2 5
2 2 3
1 0

Sample Output 3

8

Solution in C (GCC 5.4.1) (judge verdict: Accepted)

#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <string.h>
#include <ctype.h>
#include <math.h>

int main(int argc, void **argv)
{
	long long n, m;
	long long k[10], s[10][10], p[10];
	char x[10];
	long long count = 0;
	long long max;

	scanf("%lld %lld", &n, &m);
	for (int i = 0; i < m; i++)
	{
		scanf("%lld", &k[i]);
		for (int j = 0; j < k[i]; j++)
		{
			scanf("%lld", &s[i][j]);
		}
	}
	for (int i = 0; i < m; i++)
	{
		scanf("%lld", &p[i]);
	}

	//============================
	max = 1 << n;
	for (int i = 0; i < max; i++)
	{
		int t = i;
		int c;
		int ans[10];

		for (int j = 0; j < 10; j++)
		{
			x[j] = t % 2;
			t /= 2;
		}

		for (int a = 0; a < m; a++)
		{
			int status = 0;
			for (int b = 0; b < k[a]; b++)
			{
				status += x[s[a][b] - 1];
			}
			if( status % 2 == p[a] )
			{
				ans[a] = 1;
			}
			else
			{
				ans[a] = 0;
			}
		}

		for (c = 0; c < m; c++)
		{
			if( ans[c] == 0 ) break;
		}
		if( c == m )
			count++;
	}

	printf("%lld\n", count);

	return 0;
}