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<vector>
using namespace std;

int main(){
    long long N, M;
    cin >> N >> M;
    long long k[M+1];
    vector<long long> s_sub;
    long long i, j;
    for(j = 0; j <= N; j++){
        s_sub.push_back(-1);
    }
    vector<vector <long long>> s;
    for(i = 0; i <= M; i++){
        s.push_back(s_sub);
    }
    for(i = 1; i <= M; i++){
        cin >> k[i];
        for(j = 1; j <= k[i]; j++){
            cin >> s[i][j];
        }
    }
    long long p[M+1];
    for(i = 1; i <= M; i++){
        cin >> p[i];
    }
    
    long long zentori = 1, bit;
    for(j = 1; j <= N; j++){
        zentori *= 2;
    }
    long long onoff[N+1];
    long long onoffsum;
    bool countif;
    long long counta = 0;
    for(bit = 0; bit < zentori; bit++){
        countif = 1;
        for(j = 1; j <= N; j++){
            if((bit >> (j-1)) & 1){
                onoff[j] = 1;
            }else{
                onoff[j] = 0;
            }
        }
        for(i = 1; i <= M; i++){
            onoffsum = 0;
            for(j = 1; j <= k[i]; j++){
                onoffsum += onoff[s[i][j]];
            }
            if(onoffsum % 2 != p[i]){
                countif = 0;
                break;
            }
        }
        if(countif){
            counta++;
        }
    }
    cout << counta << endl;
    system("pause");
    return 0;
}