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;

class Program
{
    static string InputPattern = "InputX";

    static List<string> GetInputList()
    {
        var WillReturn = new List<string>();

        if (InputPattern == "Input1") {
            WillReturn.Add("2 2");
            WillReturn.Add("2 1 2");
            WillReturn.Add("1 2");
            WillReturn.Add("0 1");
            //1
        }
        else if (InputPattern == "Input2") {
            WillReturn.Add("2 2");
            WillReturn.Add("2 1 2");
            WillReturn.Add("1 1");
            WillReturn.Add("1 2");
            WillReturn.Add("0 0 1");
            //0
        }
        else if (InputPattern == "Input3") {
            WillReturn.Add("5 2");
            WillReturn.Add("3 1 2 5");
            WillReturn.Add("2 2 3");
            WillReturn.Add("1 0");
            //8
        }
        else {
            string wkStr;
            while ((wkStr = Console.ReadLine()) != null) WillReturn.Add(wkStr);
        }
        return WillReturn;
    }

    static void Main()
    {
        List<string> InputList = GetInputList();

        int[] wkArr = InputList[0].Split(' ').Select(X => int.Parse(X)).ToArray();
        int N = wkArr[0];
        int M = wkArr[1];

        var KArrList = new List<int[]>();
        foreach (string EachStr in InputList.Skip(1).Take(M)) {
            KArrList.Add(EachStr.Split(' ').Select(X => int.Parse(X)).ToArray());
        }
        int[] PArr = InputList[M + 1].Split(' ').Select(X => int.Parse(X)).ToArray();

        int Answer = 0;
        foreach (bool[] EachOnOffArr in DeriveOnOffArrList(N)) {
            bool IsAnswer = true;
            for (int I = 0; I <= KArrList.Count - 1; I++) {
                int SumCnt = 0;
                foreach (int EachInd in KArrList[I]) {
                    if (EachOnOffArr[EachInd - 1]) SumCnt++;
                }
                if (SumCnt % 2 != PArr[I]) {
                    IsAnswer = false;
                }
            }
            if (IsAnswer) Answer++;
        }
        Console.WriteLine(Answer);
    }

    struct JyoutaiDef
    {
        internal int CurrInd;
        internal bool[] OnOffArr;
    }

    //Nを引数として、OnOffの配列のListを返す
    static List<bool[]> DeriveOnOffArrList(int pN)
    {
        int UB = pN - 1;

        var Stk = new Stack<JyoutaiDef>();

        JyoutaiDef WillPush;
        WillPush.CurrInd = 0;
        WillPush.OnOffArr = new bool[UB + 1];
        Stk.Push(WillPush);

        var WillReturn = new List<bool[]>();
        while (Stk.Count > 0) {
            JyoutaiDef Popped = Stk.Pop();

            if (Popped.CurrInd > UB) {
                WillReturn.Add(Popped.OnOffArr);
                continue;
            }

            WillPush.CurrInd = Popped.CurrInd + 1;

            WillPush.OnOffArr = (bool[])Popped.OnOffArr.Clone();
            WillPush.OnOffArr[Popped.CurrInd] = false;
            Stk.Push(WillPush);

            WillPush.OnOffArr = (bool[])Popped.OnOffArr.Clone();
            WillPush.OnOffArr[Popped.CurrInd] = true;
            Stk.Push(WillPush);
        }

        return WillReturn;
    }
}