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;

public class App
{
    public static void Main(string[] args)
    {
        int n,m;
        {
            var r = Console.ReadLine().Split();
            n=int.Parse(r[0]);
            m=int.Parse(r[1]);  
        }
            ToggleSwitch[] Switches = new ToggleSwitch[n];
            LightBulb[] LightBulbs = new LightBulb[m];
            for(int i = 0;i < n;i++)
            {
                Switches[i] = new ToggleSwitch(false);
            }
            for(int i = 0;i < m;i++)
            {
                LightBulbs[i] = new LightBulb(Console.ReadLine().Split().Skip(1).Select(v=>Switches[int.Parse(v)-1]).ToArray(),0);
            }
            {
                var r = Console.ReadLine().Split();
                for(int i = 0;i < m;i++)
                {
                    LightBulbs[i].LitConditionModulo = int.Parse(r[i]);
                }
            }
        int PatternCount = 0;
        for(int i = 0;i < 1<<n;i++)
        {
            for(int j = 0;j < n;j++)
            {
                Switches[j].IsON = (i>>j)%2==1;
            }
            if(LightBulbs.All(v=>v.GetLit()))
                PatternCount++;
        }
        Console.WriteLine(PatternCount);
    }
    public class ToggleSwitch
    {
        public ToggleSwitch(bool InitSwitchCondition)
        {
            IsON = InitSwitchCondition;
        }
        public bool IsON;
        public void Invert()
        {
            IsON = !IsON;
        }
    }
    public class LightBulb
    {
        public readonly ToggleSwitch[] DependingSwitch;
        public int LitConditionModulo;
        public LightBulb(ToggleSwitch[] DependingSwitch,int LitConditionModulo)
        {
            if(LitConditionModulo != 0 && LitConditionModulo != 1)
                throw new InvalidOperationException("LitConditionModulo must be 0 or 1");
            this.DependingSwitch = DependingSwitch;
            this.LitConditionModulo = LitConditionModulo;
        }
        public bool GetLit()
        {
            return DependingSwitch.Count(v=>v.IsON)%2==LitConditionModulo;
        }
    }
}