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 C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Linq;

namespace AtCoder {
	class Program {
		public static void Main() {
			int[] input = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
			int n = input[0]; int m = input[1];
			int[][] s = new int[m][];
			for (int i = 0; i < m; i++) {
				s[i] = Console.ReadLine().Split(' ').Select(int.Parse).Skip(1).ToArray();
			}
			int[] p = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
			Console.WriteLine(Count(n, m, s, p, new bool[0]));

		}
		static int Count(int n, int m, int[][] s, int[] p, bool[] sw) {
			int result;
			if (sw.Length == n) {
				result = 0;
				int count = s.Where((x, i) => x.Where((_, j) => sw[j]).Count() % 2 == p[i]).Count();
				return count == m ? 1 : 0;
			}
			bool[] nsw = new bool[sw.Length + 1]; Array.Copy(sw, nsw, sw.Length);
			nsw[nsw.Length - 1] = true;
			result = Count(n, m, s, p, nsw);
			nsw[nsw.Length - 1] = false;
			result += Count(n, m, s, p, nsw);
			return result;
		}
	}
}