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.Linq;

namespace Switches
{
    class MainClass
    {
        public static void Main(string[] args)
        {
            var inputs = Console.ReadLine().Split(' ');

            var switchCount = int.Parse(inputs[0]);
            var lightCount = int.Parse(inputs[1]);

            var switchPatern = new List<SwitchHolder>();
            for (int i = int.Parse(Math.Pow(2, switchCount).ToString()) - 1; i >= 0; i--)
            {
                switchPatern.Add(new SwitchHolder(i, switchCount));
            }

            var lights = new List<Light>();
            for (int i = 0; i < lightCount; i++)
            {
                inputs = Console.ReadLine().Split(' ');
                var switchNums = new int[int.Parse(inputs[0])];
                for (int j = 1; j <= inputs.Length - 1; j++)
                {
                    switchNums[j - 1] = int.Parse(inputs[j]);
                }
                lights.Add(new Light
                {
                    CheckSwitchNumbers = switchNums.ToArray(),
                    ID = i + 1
                });
            }

            inputs = Console.ReadLine().Split(' ');
            for (int i = 0; i < lightCount; i++)
            {
                lights.ElementAt(i).Condition = int.Parse(inputs[i]);
            }

            var allLightONCount = 0;
            foreach (var patern in switchPatern)
            {
                
                var lightOnCount = 0;
                foreach (var light in lights)
                {
                    
                    if (light.IsLightON(patern.Switches))
                    {
                        lightOnCount++;
                        
                    }
                    
                }
                if (lightOnCount == lightCount)
                {
                    allLightONCount++;
                    
                }
                
                
            }

            Console.WriteLine(allLightONCount);
        }

        class Switch
        {
            public int ID { get; set; }
            public bool isON { get; set; }
        }

        class Light
        {
            public int ID { get; set; }
            public int Condition { get; set; }
            public int[] CheckSwitchNumbers { get; set; }

            public bool IsLightON(List<Switch> SwitchePatern)
            {
                var checkSwitchList = new List<Switch>();
                var switchArray = SwitchePatern.ToArray();
                for (int i = 0; i < CheckSwitchNumbers.Length; i++)
                {
                    checkSwitchList.Add(switchArray[CheckSwitchNumbers[i] - 1]);
                }

                

                var onCount = 0;
                foreach (var item in checkSwitchList)
                {
                    if (item.isON)
                    {
                        onCount++;
                    }
                }

                return onCount != 0 ? onCount % 2 == Condition : false;
            }

            public void Print()
            {
                var s = "";
                foreach (var item in CheckSwitchNumbers)
                {
                    s = s + item.ToString();
                }
                Console.WriteLine("Light ID:" + ID + " condition:" + Condition + " switch:" + s);
            }
        }

        class SwitchHolder
        {
            public List<Switch> Switches { get; set; } = new List<Switch>();

            public SwitchHolder(int a, int digits)
            {
                var bitStr = Convert.ToString(a, 2).PadLeft(digits, '0').ToCharArray();
                for (int i = 0; i < bitStr.Length; i++)
                {
                    var b = false;
                    if (bitStr[i] == '1')
                    {
                        b = true;
                    }
                    Switches.Add(new Switch { ID = i + 1, isON = b });
                }
            }

            public void Print()
            {
                var s = "";
                foreach (var item in Switches)
                {
                    if (item.isON)
                    {
                        s = s + "1";
                    }
                    else
                    {
                        s = s + "0";
                    }
                }
                Console.WriteLine("Ptn:" + s);
            }
        }
    }
}