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.ArrayList;
import java.util.HashSet;
import java.util.List;
import java.util.Scanner;
import java.util.Set;

public class Main {

    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int n = sc.nextInt();
        int m = sc.nextInt();
        List<List<Integer>> sLists = new ArrayList<>();
        int[] p = new int[m];

        for (int i = 0; i < m; i++) {
            int k = sc.nextInt();
            List<Integer> list = new ArrayList<>();
            for (int j = 0; j < k; j++) {
                list.add(sc.nextInt());
            }
            sLists.add(list);
        }
        for (int i = 0; i < m; i++) {
            p[i] = sc.nextInt();
        }

        int ans = 0;
        for (int bit = 0; bit < (1 << n); bit++) {
            Set<Integer> onOffPatter = new HashSet<>();
            for (int i = 0; i < n; i++) {
                if ((1 & (bit >> i)) == 1) {
                    onOffPatter.add(i + 1);
                }
            }

            boolean allLights = true;
            for (int i = 0; i < m; i++) {
                List<Integer> sList = sLists.get(i);
                int count = 0;
                for (int s : sList) {
                    if (onOffPatter.contains(s)) {
                        count++;
                    }
                }
                if (count % 2 != p[i]) {
                    allLights = false;
                    break;
                }
            }
            if (allLights) {
                ans++;
            }
        }
        System.out.println(ans);
    }

}