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: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;

namespace ABC128
{
	class Program
	{
		static void Main(string[] args)
		{
			var nm = Console.ReadLine().Split(' ').Select(int.Parse).ToList();
			var N = nm[0];
			var M = nm[1];
			var skm = ReadLines(M)
					.Select(x => x.Split(' ').Skip(1).Select(int.Parse).Select(i => i - 1))
					.Select((x, i) => new { k = i, s = x })
					.SelectMany(x => x.s.Select(s => new { s, k = x.k }))
					.ToList();
			var pm = Console.ReadLine().Replace(" ", "");

			var ans = 0;
			foreach (var i in Enumerable.Range(0, (int)Math.Pow(2, N)))
			{
				var sw = Convert.ToString(i, 2).Reverse();
				var rp = new int[M];

				foreach (var cs in sw.Select((c, s) => new { c, s }).Where(x => x.c == '1'))
				{
					foreach (var sk in skm.Where(x => x.s == cs.s))
					{
						rp[sk.k]++;
					}
				}

				var rpi = string.Join("", rp.Select(x => x % 2));
				if (rpi == pm) ans++;
			}
			Console.WriteLine(ans);
		}

		static IEnumerable<string> ReadLines(int n)
		{
			foreach (var _ in Enumerable.Range(0, n)) yield return Console.ReadLine();
		}
	}
}