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.*;

class light{
	int[] switches;
	int p;
	
	public light(int[] switches){
		this.switches = switches;
		this.p = 0;
	}
	public void setp(int p){
		this.p = p;
	}
}

public class Main{
	public static void main(String[] args){
		int ans = 0;
		Scanner sc = new Scanner(System.in);
		int n = sc.nextInt();
		int m = sc.nextInt();
		light[] ls = new light[m];
		
		for(int i=0; i<m; i++){
			int k = sc.nextInt();
			int[] switches = new int[k];
			for(int j=0; j<k; j++){
				switches[j] = sc.nextInt();
			}
			ls[i] = new light(switches);
		}
		
		for(int i=0; i<m; i++){
			ls[i].setp(sc.nextInt());
		}
		
		int patterns = 1;
		for(int i=0; i<n; i++){
			patterns *= 2;
		}
		
		int[] on_off = new int[n];
		for(int i=0; i<patterns; i++){
			int p_copy = i;
			for(int j=0; j<n; j++){
				on_off[j] = p_copy % 2;
				p_copy = (p_copy - p_copy%2) / 2;
			}
			int all_ok = 1;
			pattern: for(int j=0; j<m; j++){
				light light = ls[j];
				int sum = 0;
				for(int k=0; k<light.switches.length; k++){
					sum += on_off[light.switches[k]-1];
				}
				if(sum % 2 != light.p){
					all_ok = 0;
					break pattern;
				}
			}
			if(all_ok == 1){
				ans++;
			}
		}
		
		System.out.println(ans);
	}
}