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{

    static int N;
    static int M;
    static int count;
    public static void main(String args[]){
        Scanner sc = new Scanner(System.in);
        N = sc.nextInt();
        M = sc.nextInt();
        int[] k = new int[M];
        int[][] s = new int[M][N];
        for(int i = 0; i < M; i++){
            for(int j = 0; j < N; j++)
                s[i][j] = -1; 
        }
        int[] p = new int[M];
        for(int i = 0; i < M; i++){
            k[i] = sc.nextInt();
            for(int j = 0; j < k[i]; j++){
                int n = sc.nextInt();
                s[i][n-1] = 0;
            }
        }
        for(int i = 0; i < M; i++)
            p[i] = sc.nextInt();
        sc.close();
        count = 0;
        int pos = 0;
        dfs(pos, s, p);
        System.out.println(count);
    }
    static void dfs(int pos, int[][] s, int[] p){
            switch0(pos, s);
            if(pos < N-1)
                dfs(pos+1, s, p);
            if(pos == N-1){
                /*for(int i = 0; i < M; i++){
                    for(int j = 0; j < N; j++)
                        System.out.print("s["+i+"]["+j+"]="+s[i][j]+" " );
                    System.out.println();
                }*/
                if(sumjudge(s,p))
                    count++;
            }
            
            switch1(pos, s);
            if(pos < N-1)
                dfs(pos+1, s, p);
            if(pos == N-1){
                /*for(int i = 0; i < M; i++){
                    for(int j = 0; j < N; j++)
                        System.out.print("s["+i+"]["+j+"]="+s[i][j]+" " );
                    System.out.println();
                }*/
                if(sumjudge(s,p))
                    count++;
            }
    }

    static void switch0(int j, int[][] s){
        for(int i = 0; i < M; i++) {
            if(s[i][j] != -1)
                s[i][j] = 0;
        }
        return;
    }
    static void switch1(int j, int[][] s){
        for(int i = 0; i < M; i++) {
            if(s[i][j] != -1)
                s[i][j] = 1;
        }
        return;
    }
    static boolean sumjudge(int[][] s, int[] p){
        for(int i = 0; i < M; i++){
            int sum = 0;
            for(int j = 0; j < N; j++){
                if(s[i][j] == 1){
                    sum += 1;
                }
            }
            if(sum % 2 != p[i])
                return false;
        }
        return true;
    }

}