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 9.2.1) (judge verdict: Accepted)

#include <iostream>
#include <vector>
#include <set>

using namespace std;

set<int> v;

int dfs(int i, int m, int **s, int *p, int n, int *k) {
    if (i == n) {
        bool flag = true;
        for (int j = 0; j < m; j++) {
            int cnt = 0;
            for (int l = 0; l < k[j]; l++) {
                if (v.find(s[j][l]) != v.end())
                    cnt++;
            }
            if (cnt % 2 != p[j])
                flag = false;
        }
        if (flag)
            return 1;
        else 
            return 0;
    }

    int ans = 0;
    v.insert(i);
    ans += dfs(i+1, m, s, p, n, k);
    v.erase(i);
    ans += dfs(i+1, m, s, p, n, k);
    return ans;
}

int main() {
    int n, m;   cin >> n >> m;
    int *k = new int[m], **s = new int*[m], *p = new int[m];
    for (int i = 0; i < m; i++) {
        cin >> k[i];
        s[i] = new int[k[i]];

        for (int j = 0; j < k[i]; j++) {
            cin >> s[i][j];
            s[i][j]--;
        }
    }

    for (int i = 0; i < m; i++)
        cin >> p[i];

    cout << dfs(0, m, s, p, n, k) << endl;

    return 0;
}