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.awt.image.AreaAveragingScaleFilter;
import java.util.ArrayList;
import java.util.List;
import java.util.Scanner;

public class Main {
    public static void main(String[] args) {
        final Scanner scanner = new Scanner(System.in);
        final int n = scanner.nextInt();
        final int m = scanner.nextInt();

        final List<boolean[]> list = new ArrayList<>();
        final boolean[] booleans = new boolean[n];
        createArray(booleans, list, 0, n - 1);
        final List<int[]> indexList = new ArrayList<>();

        for (int i = 0; i < m; i++) {
            final int[] ints = new int[scanner.nextInt()];
            for (int j = 0; j < ints.length; j++) {
                ints[j] = scanner.nextInt() - 1;
            }
            indexList.add(ints);
        }

        for(int i = 0;i<m;i++){
            int zeroOrOne = scanner.nextInt();
            final int[] ints = indexList.get(i);

            for (int i1 = 0; i1 < list.size(); i1++) {
                final boolean[] booleans1 = list.get(i1);
                int counter = 0;
                for (final int anInt : ints) {
                    if (booleans1[anInt]) {
                        counter++;
                    }
                }
                if (counter % 2 != zeroOrOne) {
                    list.remove(i1);
                    i1--;
                }
            }
        }


        System.out.println(list.size());
    }

    private static void createArray(final boolean[] bools, final List<boolean[]> list, final int index, final int last) {
        if (index == last) {
            bools[index] = true;
            list.add(getNewArray(bools));
            bools[index] = false;
            list.add(getNewArray(bools));
            return;
        }

        bools[index] = true;
        createArray(bools, list, index + 1, last);
        bools[index] = false;
        createArray(bools, list, index + 1, last);
    }

    private static boolean[] getNewArray(final boolean[] original) {
        final boolean[] booleans = new boolean[original.length];
        for (int i = 0; i < original.length; i++) {
            booleans[i] = original[i];
        }
        return booleans;
    }
}