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 {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int n = sc.nextInt();
        int m = sc.nextInt();
        HashMap<Integer, ArrayList<Integer>> map = new HashMap<>();
        int[] p = new int[m];
        for(int i = 0; i < m; i++) {
            int k = sc.nextInt();
            ArrayList<Integer> list = new ArrayList<>();
            for(int j = 0; j < k; j++) {
                list.add(sc.nextInt());
            }
            map.put(i, list);
        }
        for(int i = 0; i < m; i++) {
            p[i] = sc.nextInt();
        }
        int num = 1 << n;
        int ans = 0;
        // numは電球のon/off
        for(int i = 0; i < num; i++) {
            ans++;
            for(Map.Entry<Integer, ArrayList<Integer>> entry: map.entrySet()) {
                int index = entry.getKey();
                ArrayList<Integer> list = entry.getValue();
                int c = count(list, i);
                if(c % 2 != p[index]) {
                    ans--;
                    break;
                }
            }
        }
        System.out.println(ans);
    }
    // いくつの電球がonかを数える
    private static int count(ArrayList<Integer> list, int num) {
        int count = 0;
        // i番目の電球を調べる
        for(int i: list) {
            i--;
            if((num & (1 << i)) != 0) count++;
        }
        return count;
    }
}