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.io.BufferedReader;
import java.io.InputStreamReader;
import java.util.HashSet;
import java.util.Iterator;
import java.util.Set;

public class Main {
    public static void main(String[] args) throws Exception {
        BufferedReader reader = new BufferedReader(new InputStreamReader(System.in));
        String[] nm = reader.readLine().split(" ");
        int n = Integer.parseInt(nm[0]);
        int m = Integer.parseInt(nm[1]);
        int[] k = new int[m];
        int[][] s = new int[m][];
        for (int i = 0; i < m; i++) {
            String[] arr = reader.readLine().split(" ");
            k[i] = Integer.parseInt(arr[0]);
            s[i] = new int[k[i]];
            for (int j = 0; j < k[i]; j++) {
                s[i][j] = Integer.parseInt(arr[j + 1]);
            }
        }
        String[] parr = reader.readLine().split(" ");
        int[] p = new int[m];
        for (int i = 0; i < m; i++) {
            p[i] = Integer.parseInt(parr[i]);
        }
        int maxState = 1 << n;
        Set<Integer> set = new HashSet<>(maxState);
        for (int i = 0; i < maxState; i++) {
            set.add(i);
        }
        for (int i = 0; i < m; i++) {
            for (Iterator<Integer> iter = set.iterator(); iter.hasNext(); ) {
                int cur = iter.next();
                int count = 0;
                for (int j = 0; j < k[i]; j++) {
                    int tmp = (cur >> (s[i][j] - 1)) & 1;
                    if (tmp == 1) {
                        count++;
                    }
                }
                if (count % 2 != p[i]) {
                    iter.remove();
                }
            }
        }
        System.out.println(set.size());
    }
}