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<bits/stdc++.h>
#define rep(i, n) for(int i=0;i<n;i++)
typedef long long ll;
using namespace std;

int main(){
    int n, m;
    cin >> n >> m;

    vector<int> k(m);
    vector<vector<int>> s(m);
    vector<int> p(m);

    rep(i, m){
        cin >> k[i];
        rep(j, k[i]){
            int tmp;
            cin >> tmp;
            s[i].push_back(tmp);
        }
    }
    rep(i, m) cin >> p[i];

    int ans = 0;

    vector<int> sw(n);

    for(int i=0;i<pow(2,n);i++){
        bool flag = 1;
        int tmp = i;

        for(int j=n-1;j>=0;j--){
            sw[j] = tmp % 2;
            tmp /= 2;
        }

        //cout << sw[0] << sw[1] << endl;

        for(int j=0;j<m;j++){
            int sum=0;
            for(int l=0;l<k[j];l++){
                sum += sw[s[j][l]-1];
                //cout << s[j][l]-1 << ": " << sw[s[j][l]-1] << endl;
            }
            //cout << sum << p[j] << endl;
            if(sum % 2 == p[j]) flag *= 1;
            else flag *= 0;
        }

        if(flag) ans++;
    }

    cout << ans << endl;

    return 0;
}