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.Scanner;

public class Main {

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

		int N = sc.nextInt();
		int M = sc.nextInt();

		int[][] S = new int[M][N+1];

		for(int i = 0 ; i < M ; i ++){
			int k = sc.nextInt();
			for(int j = 0 ; j < k ; j++){
				S[i][j] = sc.nextInt();
			}
		}

		int[] P = new int[M];
		for(int i = 0 ; i < M ; i++){
			P[i] = sc.nextInt();
		}
		sc.close();

		//スイッチパターンを2進数で表現
		int aans = (int)Math.pow(2, N);

		String denkyu;
		//スイッチオフを表現するために、桁埋め
		String ar = "%0" + N + "d";

		int count = 0;
		int a = 0;
		int ans = 0;
		for(int i = 0 ; i < aans ; i++){
			denkyu = String.format(ar ,Integer.parseInt(Integer.toBinaryString(i)));
			//System.out.println(denkyu);


			//各電球ごとのスイッチパターンを全探索
			for(int j = 0 ; j < M ; j++){
				for(int k = 0 ; k < N ; k++){
					if(S[j][k] ==0){
						break;
					}
					//System.out.println("S : " + S[j][k]);

					//スイッチオンの数を合計
					a += Integer.parseInt(String.valueOf(denkyu).substring(S[j][k]-1,S[j][k]));

					}
				//System.out.println("a : " + a);

				//2で割ったときの余りは適切か
				if(a%2 == P[j]){
					count++;
				}
				a = 0;
			}
			//System.out.println(count);
			//全電球が点灯するパターン(countがMになるパターン)は何個あるか
			if(count == M){
				ans++;
			}
			count = 0;
		}

		System.out.println(ans);
	}

}