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.HashSet;
import java.util.LinkedHashMap;
import java.util.Map;
import java.util.Scanner;
import java.util.Set;

public class Main {
	public static void main(String[] args) {
		Scanner stdIn = new Scanner(System.in);

		int numberOfSwitch = stdIn.nextInt();
		int numberOfLights = stdIn.nextInt();

		stdIn.nextLine();//改行のため

		Map<Integer, LightSwitch> switchMap = new LinkedHashMap<>();
		// set input
		for (int i = 1; i <= numberOfLights; i++) {
			switchMap.put(i, new LightSwitch(i, stdIn.nextLine()));
		}
		for (int i = 1; i <= numberOfLights; i++) {
			switchMap.get(i).setMod(stdIn.nextInt());
		}

		// execute
		int allSize = (int) Math.pow(2, numberOfSwitch);
		int cnt = 0;

		for (int i = 0; i < allSize; i++) {

			String boolStr = String.format("%" + numberOfSwitch + "s", Integer.toBinaryString(i)).replaceAll(" ", "0");

			if (canAllLightUp(switchMap, boolStr)) {
				cnt++;
			}
		}

		System.out.println(cnt);

		stdIn.close();
	}

	private static boolean canAllLightUp(Map<Integer, LightSwitch> switchMap, String boolStr) {
		for (LightSwitch swt : switchMap.values()) {
			if (!swt.canLightsUp(boolStr)) {
				return false;
			}
		}

		return true;
	}
}

class LightSwitch {
	private int switchId;
	private Set<Integer> useSwitchSet = new HashSet<>();
	private int mod;

	public LightSwitch(int switchId, String useSwitchStr) {
		this.switchId = switchId;

		String[] strArray = useSwitchStr.split(" ");

		for (int i = 1; i < strArray.length; i++) {
			useSwitchSet.add(Integer.parseInt(strArray[i]));
		}
	}

	public boolean canLightsUp(String switchArrayStr) {
		int cnt = 0;

		for (Integer switchId : useSwitchSet) {
			if (switchArrayStr.charAt(switchId - 1) == '1') {
				cnt++;
			}
		}

		return cnt % 2 == mod;

	}

	public int getMod() {
		return mod;
	}

	public void setMod(int mod) {
		this.mod = mod;
	}

	public int getSwitchId() {
		return switchId;
	}

}