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 int N;
    public static int M;
    public static List<Integer> k;
    public static List<List<Integer>> s;
    public static List<Integer> p;
    public static void main (String args[]){
        Scanner scanner = new Scanner(System.in);
        N = scanner.nextInt();
        M = scanner.nextInt();
        k = new ArrayList();
        s = new ArrayList();
        p = new ArrayList();
        for(int i=0; i<M; i++){
            int ki = scanner.nextInt();
            k.add(ki);
            List<Integer> si = new ArrayList();
            for(int j=0; j<ki; j++){
                si.add(scanner.nextInt());
            }
            s.add(si);
        }
        for(int i=0; i<M; i++){
            p.add(scanner.nextInt());
        }
        int numOK = 0;
        List<Integer> switches = new ArrayList();
        Queue<List<Integer>> queue = new ArrayDeque();
        List<Integer> states = new ArrayList(Arrays.asList(0,1));
        queue.add(switches);
        while( queue.size()>0 ){
            List<Integer> now = queue.poll();

            if(now.size() == N){
                int numOnBulb = 0;
                for(int i=0; i<M; i++){
                    int numOnSwitch = 0;
                    for(Integer sij:s.get(i)){
                        numOnSwitch = numOnSwitch + now.get(sij-1);
                    }
                    if(numOnSwitch%2 == p.get(i)){
                        numOnBulb++;
                    }
                }
                if(numOnBulb==M){
                    numOK++;
                }
            }
            else{
                for(Integer nextState:states){
                    List<Integer> next = new ArrayList(now);
                    next.add(nextState);
                    if(isOK(next)){
                        queue.offer(next);
                    }
                }
            }
        }
        System.out.println(numOK);
    }

    public static boolean isOK(List<Integer> next){
        // for(int i=0; i<M; i++){
        //     int maxS = Math.max(s.get(i));
        //     if(maxS <= next.size()){
        //         int sum = 0;
        //         for(int state:next){
        //             sum = sum + state;
        //         }
        //         if()
        //         return false;
        //     }
        // }
        return true;
    }
}