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: Wrong Answer)

import java.util.*;
import java.util.stream.Collectors;
//import java.io.*;
import java.util.stream.IntStream;


public class Main{
    static int sum=0;
    public static void main (String args[]){
        //初期処理
        Scanner sc=new Scanner(System.in);
        // BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        // try{
            // int n = Integer.parseInt(br.readLine());
            int N= sc.nextInt();
            int M=sc.nextInt();
            Bulb[] bulbs = new Bulb[M];
            int[] swArr = new int[N];
            Set<int[]> swSet = new HashSet<>();

            setSwPattern(swSet, swArr, 0);


            IntStream.range(0,M).forEach(i->{
                int n=sc.nextInt();
                bulbs[i]=new Bulb();
                for(int j=0;j<n;j++){
                    int m=sc.nextInt();
                    bulbs[i].sw.add(m);
                }
            });

            IntStream.range(0,M).forEach(i->bulbs[i].p=sc.nextInt());

            swSet.forEach(s->check(s,bulbs));
            System.out.println(sum);

    }

    static void check(int[] swArr, Bulb[] bulbs){

            Arrays.asList(bulbs).stream().forEach(s->{
                int swSum=0;
                for(int i=0;i<s.sw.size();i++){
                    swSum+=swArr[i];
                }
                if(s.p==(swSum%2)){
                    s.on=true;
                }else{
                    s.on=false;
                }
            });
            if(Arrays.asList(bulbs).stream().allMatch(s->s.on==true)){
                sum++;
            }
        
    }
    static void setSwPattern(Set<int[]> set,int[] swArr,int n){
        set.add(swArr);
        if(n<swArr.length){
        int cloneSwArr[]= swArr.clone();
        swArr[n]=0;
        setSwPattern(set,swArr,n+1);
        cloneSwArr[n]=1;
        setSwPattern(set,cloneSwArr,n+1);
        }
    }

    
}


class Bulb{
    Bulb(){};
    int p;
    Boolean on=false;
    List<Integer> sw = new ArrayList<>();

    void turn(){
        this.on=(this.sw.stream().reduce(0,Integer::sum)/2)==this.p;
    }

}