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 5.4.1) (judge verdict: Accepted)

#include<stdio.h>
int sw_state[11],sw_ans[11],sig[11];
int sw[11][11];
int sw_n,vol_n,ans;
void check(int depth);
int main(void){
    int i,j,k,temp;
    scanf("%d %d",&sw_n,&vol_n);
    for(i=0;i<11;i++){
        for(j=0;j<11;j++){
            sw[i][j]=0;
        }
    }
    for(i=1;i<=vol_n;i++){
        scanf("%d",&k);
        for(j=0;j<k;j++){
            scanf("%d",&temp);
            sw[temp][i]=1;
        }
    }
    for(i=1;i<=vol_n;i++){
        scanf("%d",&sw_ans[i]);
    }
    ans=0;
    check(1);
    printf("%d\n",ans);
    return 0;
}
void check(int depth){
    int i,j,fg;
    if(depth>sw_n){
        for(i=0;i<11;i++){
            sig[i]=0;
        }
        for(i=1;i<=sw_n;i++){
            for(j=1;j<=vol_n;j++){
                sig[j]+=sw[i][j]*sw_state[i];
            }
        }
        fg=1;
        for(i=1;i<=vol_n;i++){
            if(sig[i]%2 != sw_ans[i]){
                fg=0;
                break;
            }
        }
        ans+=fg;
    }else{
        sw_state[depth]=0;
        check(depth+1);
        sw_state[depth]=1;
        check(depth+1);
    }
    return;
}