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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.util.*;

class Main{
  public static void main(String[] args){
    Scanner sc=new Scanner(System.in);
    int n=sc.nextInt();
    int m=sc.nextInt();
    int[][] a=new int[m][];
    for(int i=0;i<m;i++){
      int k=sc.nextInt();
      a[i]=new int[k];
      for(int j=0;j<k;j++){
        a[i][j]=sc.nextInt()-1;
      }
    }
    int[] p=new int[m];
    for(int i=0;i<m;i++){
      p[i]=sc.nextInt();
    }
    int ans=0;
    Deque<E> dq=new ArrayDeque<>();
    dq.add(new E(n));
    while(!dq.isEmpty()){
      E here=dq.pollFirst();
      if(here.fixed<n){
        dq.add(here.nT());
        dq.add(here.nF());
        continue;
      }
//      System.err.println(Arrays.toString(here.sw));
      boolean flag=true;
      for(int i=0;i<m;i++){
        int cnt=0;
        for(int j=0;j<a[i].length;j++){
          cnt+=here.sw[a[i][j]]?1:0;
        }
        if((cnt&1)!=p[i]){
          flag=false;
          break;
        }
      }
      if(flag){
        ans++;
      }
    }
    System.out.println(ans);
  }
  static class E{
    boolean[] sw;
    int fixed=0;
    E(int n){
      sw=new boolean[n];
    }
    E nT(){
      E res=new E(sw.length);
      for(int i=0;i<fixed;i++){
        res.sw[i]=sw[i];
      }
      res.sw[fixed]=true;
      res.fixed=fixed+1;
      return res;
    }
    E nF(){
      E res=new E(sw.length);
      for(int i=0;i<fixed;i++){
        res.sw[i]=sw[i];
      }
      res.fixed=fixed+1;
      return res;
    }
  }
}