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.*;

public class Main {
    private static int[] vim;
    private static boolean[] flag;
    private static List<Buld> list;
    private static int sum;
    public static void main(String[] args) {
        Scanner in = new Scanner(System.in);
        int n = in.nextInt();
        int m = in.nextInt();
        list = new ArrayList<>();
        vim = new int[n];
        flag = new boolean[n];
        sum = 0;
        for(int i=0;i<m;i++){
            int k = in.nextInt();
            List<Integer> swi = new ArrayList<>();
            for(int j=0;j<k;j++){
                swi.add(in.nextInt());
            }
            list.add(new Buld(swi));
        }
        for(int i=0;i<m;i++){
            list.get(i).p=in.nextInt();
        }
        dfs(0,n,m);
        System.out.println(sum);
    }
    public static void dfs(int x,int n,int m){
        if(x == n){
            for(int i=0;i<m;i++){
                Buld b = list.get(i);
                int att=0;
                for(Integer s:b.swi){
                    att^=vim[s-1];
                }
                if(att != b.p)
                    return;
            }
            sum++;
        } else {
            for (int i = 0; i < 2; i++) {
                vim[x] = i;
                dfs(x + 1, n, m);
            }
        }
    }
    static class Buld{
        public List<Integer> swi;
        public Integer p;
        public Buld(List<Integer> list){
            this.swi = list;
        }

    }
}