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 <algorithm>

using namespace std;

int main(void){
    int N,M; cin >> N >> M;
    vector<int> k(M),p(M);
    vector<vector<int>> s(M,vector<int>(N));
    for(int i=0;i<M;i++){
        cin >> k[i];
        for(int j=0;j<k[i];j++){
            cin >> s.at(i).at(j);
        }
    }
    for(int i=0;i<M;i++){
        cin >> p[i];
    }
    int ans = 0;
    vector<int> b(N+1);//binary number
                       //スイッチのオンオフを0か1かでM+1個定義
                       //M+1が１、すなわち1000...0になったら全通り試したので停止
    int a=0;
    while(1){
        for(int i=0;i<N;i++){
            if(b[i]==2){
                b[i] = 0;
                b[i+1]++;
            }
        }
        //以上がN個のスイッチのオンオフ切り替えの全探索
        
        if(b[N]==1)break;

        for(int i=0;i<M;i++){
            int count = 0;
            for(int j=0;j<k[i];j++){
                if(b[s.at(i).at(j)-1]==1){//「-1」は実際の入力は1からであるため
                    count++;
                }
            }                
            if(count%2!=p[i])break;
            //以上がM個ある電球のうち電球iの必要スイッチと実際のスイッチの照らし合わせ

            if(i==M-1){
                ans++;
            }
            //以上が、とあるスイッチオンオフの組み合わせにおける成功の可否
        }
        b[0]++;
    }
    cout << ans << endl;
}