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) throws Exception {
        Scanner sc = new Scanner(System.in);
        
        int switch_n = Integer.parseInt(sc.next());
        int light_n = Integer.parseInt(sc.next());
        
        boolean[][] sw_to_light = new boolean[switch_n][light_n];
        
        for(int i=0;i<switch_n;i++){
            for(int j=0;j<light_n;j++){
                sw_to_light[i][j]=false;
            }
        }
        
        int connected_switch=0;
        int to_switch=0;
        for(int i=0;i<light_n;i++){
            connected_switch = Integer.parseInt(sc.next());
            for(int j=0;j<connected_switch;j++){
                to_switch = Integer.parseInt(sc.next());
                sw_to_light[to_switch-1][i] = true;
            }
        }
        
        boolean[] when_lit=new boolean[light_n];
        
        int tmp=0;
        for(int i=0;i<light_n;i++){
            tmp = Integer.parseInt(sc.next());
            if(tmp==1){
                when_lit[i]=true;
            }else{
                when_lit[i]=false;
            }
        }
        boolean[] light_state=new boolean[light_n];
        
        boolean all_lit=true;
        int count=0;
        
        for(int i=0;i <(1<<switch_n);i++){
            for(int j=0;j<light_n;j++){
                light_state[j]=false;
            }   //Initialization
            
            for(int j=0;j<switch_n;j++){
                if((i&(1<<j))>0){
                    for(int k=0;k<light_n;k++){
                        if(sw_to_light[j][k]) light_state[k]=!light_state[k];
                    }
                }
            }
            
            all_lit=true;
            for(int j=0;j <light_n;j++){
                if(light_state[j] != when_lit[j]) all_lit=false;
            }
            if(all_lit) count++;
        }
        
        //String s = sc.next();
        
        System.out.println(count);
    }
}