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 <iostream>
#include <string>
#include <vector>
#include <utility>
#include <cmath>

using namespace std;

int main(){
    int n, m;
    cin >> n >> m;
    vector<int> k(m), p(m);
    vector< vector<int> > s(m, vector<int>(0));
    for(int i=0;i<m;i++){
        cin >> k.at(i);
        for(int j=0;j<k.at(i);j++){
            int tmp;
            cin >> tmp;
            s.at(i).push_back(tmp);
        }
    }
    for(int i=0;i<m;i++){
        cin >> p.at(i);
    }

    int cnt = 0;
    int pattern_max = pow(2, n) - 1;
    vector<bool> swit(n, false);

    for(int i=0;i<=pattern_max;i++){
        int pnum = i;
        for(int j=0;j<n;j++){
            if(pnum >= pow(2, (n-1)-j)){
                swit.at((n-1)-j) = true;
                pnum -= pow(2, (n-1)-j);
            }
        }

        bool flag = true;
        for(int j=0;j<m;j++){
            int s_cnt = 0;
            for(int l=0;l<k.at(j);l++){
                if(swit.at(s.at(j).at(l)-1))s_cnt++;
            }
            if(s_cnt%2 != p.at(j))flag = false;
        }
        if(flag)cnt++;
        for(int j=0;j<n;j++)swit.at(j) = false;
    }

    cout << cnt << endl;
}