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 static Exter;
using System.Linq;
#if DEBUG
#else
using console = System.Console;
#endif

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;
        }
    }
}
public static class Exter
{
    public class Point
    {
        public Point(int cx, int cy)
        {
            x = cx;
            y = cy;
        }
        public Point(long[] carr)
        {
            x = carr[0].ToInt();
            y = carr[1].ToInt();
        }
        public void Swap()
        {
            int p = x;
            x = y;
            y = p;
        }
        public int x;
        public int y;
    }
    public static void Rep(int loop,Action<int> act)
    {
        for(int i=0;i<loop;i++)
            act(i);
    }
    public static IEnumerable<Result> ListAll<T1,T2,Result>(IEnumerable<T1> a,IEnumerable<T2> b,Func<T1,T2,Result> f)
    {
        foreach(var v in a)
        {
            foreach(var w in b)
               yield return f(v,w);
        }
    }
    public static IEnumerable<Result> ListAll<T1,T2,Result>(IEnumerable<T1> a,IEnumerable<T2> b,Func<T1,T2,Result> f,Func<T1,T2,bool> s)
    {
        foreach(var v in a)
        {
            foreach(var w in b)
               if(s(v,w)) yield return f(v,w);
        }
    }
    public static int ToInt(this long v)=>(int)v;
    public static int ToInt(this string v)=>int.Parse(v);
    public static int Max(params int[] p)=>p.Max();
    public static long Max(params long[] p)=>p.Max();
    public static int Min(params int[] p)=>p.Min();
    public static long Min(params long[] p)=>p.Min();
    public static int Gcd(int a, int b)=>a > b ? GcdRecursive(a, b) : GcdRecursive(b, a);
    private static int GcdRecursive(int a, int b)=>b == 0 ? a : GcdRecursive(b, a % b);
    public static object cout
    {
        set
        {
            console.Write(value.ToString());
            console.WriteLine();
            
        }
    }
    public static void ToCout<t>(this IEnumerable<t> v)
    {
        cout = string.Join(",",v);
    }
    public static long[] readarr
    {
        get 
        {
            preadarr = console.ReadLine().Split().Select(s=>long.Parse(s)).ToArray();
            return preadarr;
        }
    }
    public static long[] preadarr;
}


#if DEBUG
public static class console
{
    static int cursorLine = 0;
    static readonly string[] cin = new string[]
    {
                "5 2",
"3 1 2 5","2 2 3",
"1 0"
    };
    static public string ReadLine()
    { 
        return cin[cursorLine++];
    }
    static public void WriteLine()
    {
        Console.WriteLine();
    }
    static public void WriteLine<T>(T input)
    {
        Console.WriteLine(input);
    }
    static public void Write<T>(T input)
    {
        Console.Write(input);
    }
    static public void WriteLine(string s,params object[] arr)
    {
        Console.WriteLine(s,arr);
    }
}
#endif