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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <cstdio>
#include <iostream>
#include <algorithm>
#include <cmath>
#include <string>
#include <vector>
#include <queue>
#include <bitset>

using namespace std;

typedef long long ll;
typedef unsigned long long ull;

int search(int N, int M, vector<vector<int>> &s, vector<int> &p) {
    int count = 0;

    for(int i=0; i<pow(2, N); i++) {
        bitset<10> b(i);
        vector<int> q(M);
        for(int j=0; j<N; j++) {
            if(b[j]==1) {
                for(int k=0; k<M; k++) {
                    q[k] += s[k][j];
                }
            }
        }

        bool f = true;
        for(int j=0; j<M; j++) {
            f = f && ((p[j]+q[j]) % 2 == 0);
        }

        if(f)
            count++;
    }

    return count;
}

int main() {
    int N, M;
    cin >> N >> M;

    vector<vector<int>> s(M);
    for(int i=0; i<M; i++) {
        int k;
        cin >> k;
        s[i] = vector<int>(N);
        for(int j=0; j<k; j++) {
            int si;
            cin >> si;
            s[i][si-1] = 1;
        }
    }

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


    auto count = search(N, M, s, p);

    cout << count << endl;

    return 0;
}