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>
using namespace std;

int main(){
    int N, M;
    int klst[10] = {0};
    int slst[10][10];
    for(int i=0; i<10; i++){
        for(int j=0; j<10; j++){
            slst[i][j] = 0;
        }
    }
    int plst[10] = {0};
    cin >> N >> M;
    for(int i=0; i<M; i++){
        cin >>klst[i];
        for(int j=0; j<klst[i]; j++){
            cin >> slst[i][j];
        }
    }
    for(int i=0;i<M;i++){
        cin >> plst[i];
    }

    int cnt = 0;
    int m = (1 << N);
    for(int i=0;i<m;i++){
        int on_switch_num[10] = {0};
        for(int j=0;j<N;j++){
            if(((i >> j) & 1) == 1){
                for(int k=0; k<M; k++){
                    for(int l=0; l<klst[k]; l++){
                        if(slst[k][l] == j+1){
                            on_switch_num[k]++;
                        }
                    }
                }
            }
        }
        bool flg = true;
        for(int i=0;i<M;i++){
            if(on_switch_num[i]%2 != plst[i]){
                flg = false;
                break;
            }
        }
        if(flg){
            cnt++;
        }
    }
    cout << cnt;
    return 0;
}