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 <algorithm>
#include <bitset>
#include <cassert>
#include <cmath>
#include <climits>
#include <cstdlib>
#include <iostream>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <string>
#include <vector>
using namespace std;

int main() {
    int N, M; cin >> N >> M;
    vector<vector<int>> data(M,vector<int>(N));
    vector<int> p(M),counter(M);
    int k = 0,answer=0;
    for (int i = 0; i < M; i++) {
        cin >> k;
        for (int j = 0; j < k; j++) {
            cin >> data.at(i).at(j);
        }
    }
    for (int i = 0; i < M; i++) {
        cin >> p.at(i);
    }

    for (int bit = 0; bit < (1 << N); ++bit) {
        
        fill(counter.begin(), counter.end(), 0);
        for (int i = 0; i < N; ++i) {
            if (bit & (1 << i)) { 
                for (int j = 0; j < M; j++) {
                    auto itr = find(data.at(j).begin(), data.at(j).end(), i+1);
                    if (itr != data.at(j).end())counter.at(j)++;
                }
            }
        }

        for (int i = 0; i < M; i++) {
            if (counter.at(i) % 2 != p.at(i))break;
            if (i == M-1)answer++;
        }
    }
    cout << answer << endl;
    return 0;
}