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 = Integer.parseInt(sc.next());
        int m = Integer.parseInt(sc.next());

        Map<Integer, List<Integer>> d = new HashMap<>();
        for (int i = 0; i < m; i++) {
            int k = Integer.parseInt(sc.next());
            for (int j = 0; j < k; j++) {
                int s = Integer.parseInt(sc.next()) - 1;
                List<Integer> l = d.getOrDefault(s, new ArrayList<>());
                l.add(i);
                d.put(s, l);
            }
        }

        int[] p = new int[m];
        for (int i = 0; i < m; i++) {
            p[i] = Integer.parseInt(sc.next());
        }

        int res = 0;

        for (int i = 0; i < (1 << n); i++) {
            int[] ii = new int[m];
            for (int j = 0; j < n; j++) {
                if((1 & i >> j) == 1) {
                    List<Integer> l = d.getOrDefault(j, new ArrayList<>());
                    for (int iii : l) {
                        ii[iii] += 1;
                    }
                }
            }

            boolean b = true;
            for (int j = 0; j < m; j++) {
                if(ii[j] % 2 != p[j]) {
                    b = false;
                    break;
                }
            }
            if(b) {
                res++;
            }
        }

        System.out.println(res);
    }
}