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>
#include <algorithm>
#include <random>
#include <string>
#include <cmath>
#include <sstream>
#include <istream>
#include <iomanip>
#include <typeinfo>
#include <stack>

using namespace std;
typedef long long ll;
ll N;

int main(){
    ll N, M;
    cin >> N >> M;

    ll K;
    vector<vector <ll > > balb;
    for(ll i = 0; i < M; i++){
        cin >> K;
        vector<ll > s(K);
        for(ll j = 0; j < K; j++){
            cin >> s[j];
        }
        balb.push_back(s);
        /*
        for(ll j = 0; j < balb[i].size(); j++){
            cout << balb[i][j] << ",";
        }
        cout << endl;
        */
    }

    vector<ll > P(M, 0);
    for(ll i = 0; i < M; i++){
        cin >> P[i];
    }

    //bit全探査
    ll sum = 0;
    for(ll bit = 0; bit < (1<<N); ++bit){
        //電球ごとに点灯するかチェック
        bool check = true;
        for(ll i = 0; i < M; i++){
            ll count = 0;
            for(ll j = 0; j < balb[i].size(); j++){
                //どのスイッチがONになっていて何個ONになっているか確認
                //cout <<balb[i][j] << endl;
                //つじつまをあわせるために-1する
                if(bit & (1<<balb[i][j]-1)){
                    count++;
                   
                }
            }
            //P次第で，偶数または奇数の場合光るかどうかを判断する
            if(count % 2 != P[i]){
                check = false;
            }

        }
        if(check == true){
            sum++;
        }
    }

    cout << sum << endl;

}