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++ (GCC 9.2.1) (judge verdict: Accepted)

#include <iostream>
#include<vector>
#include<bitset>
#define rep(i,n) for(int i = 0; i < (n); i++)

using namespace std;



int main() {
    //入力部
    int n,m;
    cin>>n>>m;
    vector<bitset<10>>a(n);     //n番目のスイッチがどの電球とつながっているかを表すビット列
    rep(i,m){
        int k;
        cin>>k;
        rep(j,k){
            int s;
            cin>>s;
            s--;
            a[s] |= 1<<i;       //電球の番号のビットを立てる
        }
    }
    bitset<10>p;        //目標のビット列
    rep(i,m){
        int x; cin>>x;
        p |= x<<i;
    }
    
    //ここから全探索
    int ans =0;     //条件を満たす方法が何通りあるかの答え
    rep(i,1<<n){        //スイッチの押し方を表すビット列で全探索
        bitset<10>t;        //現在の電球の状態を示すビット列
        rep(j,n){
            if((i>>j)&1){
                t ^= a[j];      //xorをとってスイッチを押す
            }
        }
        if(t == p){
            ans++;
        }
	}

    cout<<ans<<endl;
	
    

    return 0;
}