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 (Clang 3.8.0) (judge verdict: Accepted)

#include <stdio.h>

int sw[16];
int n, m;
int s[16][16];
int k[16];
int p[16];

int dfs(int pos);

int main()
{
    scanf("%d %d", &n, &m);
    for (int i = 0; i < m; i++) {
        scanf("%d", &k[i]);

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

    printf("%d\n", dfs(0));

    return 0;
}

int dfs(int pos)
{
    if (pos == n) {
        for (int i = 0; i < m; i++) {
            int sum = 0;
            for (int j = 0; j < k[i]; j++) {
                sum += sw[s[i][j]];
            }

            if (sum % 2 != p[i]) {
                return 0;
            }
        }

        return 1;
    }
    
    int res = 0;

    sw[pos] = 1;
    res += dfs(pos + 1);
    sw[pos] = 0;
    res += dfs(pos + 1);

    return res;
}