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.Collections.Generic;
using System.Text;

namespace ConsoleApplication1
{
	class Program
	{
		static void Main(string[] args)
		{
			var input = Console.ReadLine().Split(' ');
			int n = int.Parse(input[0]);    // スイッチの数
			int m = int.Parse(input[1]);    // 電球の数

			var lights = new List<Light>(m);
			for (int i = 0; i < m; i++)
			{
				var arg = Console.ReadLine();
				lights.Add(new Light(n, arg));
			}
			var rems = Console.ReadLine().Split(' ');
			for (int i = 0; i < m; i++)
			{
				lights[i].SetRemainder(int.Parse(rems[i]));
			}

			int maxSwitchPattern = (int)Math.Pow(2, n) - 1;
			int countAllLightOn = 0;
			for (int i = 0; i <= maxSwitchPattern; i++)
			{
				bool isLightOn = true;
				// 各電球について確認
				for (int j = 0; j < m; j++)
				{
					if (!lights[j].IsLightOn(i))
					{
						isLightOn = false;
						break;
					}
				}
				if (isLightOn)
				{
					countAllLightOn++;
				}
			}

			Console.WriteLine(countAllLightOn);
		}

		public class Light
		{
			int mTotalSw = 0;
			int mRemainder = 0;
			int mSwitch = 0;

			public Light(int totalSw, string input)
			{
				mTotalSw = totalSw;
				var args = input.Split(' ');
				int switchCount = int.Parse(args[0]);
				var switchIndex = new List<int>(switchCount);
				for (int i = 0; i < switchCount; i++)
				{
					switchIndex.Add(int.Parse(args[i + 1]) - 1);
				}
				for (int i = 0; i < mTotalSw; i++)
				{
					mSwitch = mSwitch << 1;
					if (switchIndex.Contains(i))
					{
						mSwitch |= 1;
					}
				}
			}

			public void SetRemainder(int remainder)
			{
				mRemainder = remainder;
			}

			public bool IsLightOn(int switchState)  // switchState : N(<=10)ビットの整数. S1, S2, .... SN の状態を表すもの。ON ならビットを立てる
			{
				if (switchState == 0)
				{
					return false;    // 全てOFFの場合は余りの数に関係なく非点灯？
				}
				var state = switchState & mSwitch;   // 関連づいていてONになっているビットが立つ
				int onSwitchCount = 0;
				for (int i = 0; i < mTotalSw; i++)
				{
					if ((state & 1) == 1)
					{
						onSwitchCount++;
					}
					state = state >> 1;
				}

				return (onSwitchCount % 2 == mRemainder);
			}
		}
	}
}