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: Runtime Error)

using System;
using System.Linq;
using System.Collections.Generic;

namespace ABC128
{
    class C
    {
        static void Main(string[] args)
        {
            string[] input = Console.ReadLine().Split(' ');
            int N = int.Parse(input[0]);
            int M = int.Parse(input[1]);

            int[] k = new int[M];
            int[,] s = new int[M, 10];

            for (int i = 0; i < M; i++)
            {
                input = Console.ReadLine().Split(' ');
                k[i] = int.Parse(input[0]);

                for (int j = 1; j <= k[i]; j++)
                {
                    s[i, j] = int.Parse(input[j]);
                    //Console.WriteLine("---");
                    //Console.WriteLine(s[i, j]);
                }
            }

            input = Console.ReadLine().Split(' ');
            int[] p = new int[M];
            for (int i = 0; i < M; i++)
            {
                p[i] = int.Parse(input[i]);
            }

            int[] a = new int[N];
            CalcSw(N, a, M, k, s, p);

            Console.WriteLine(count);

        }

        static int count = 0;
        static void CalcSw(int N, int[] a, int M, int[] k, int[,] s, int[] p)
        {
            if (N == 0)
            {
                // foreach (var x in a) { Console.Write(x); }
                // Console.WriteLine();

                // 一番下まできたら、Mとｋでループして、
                // 電気がついているかどうかを確認する
                // 計算したらメモ化再帰のでカウントする
                int c = 0;
                for (int i = 0; i < M; i++)
                {
                    c = 0;
                    //Console.WriteLine("------");
                    for (int j = 1; j <= k[i]; j++)
                    {
                        //Console.WriteLine(s[i, j]);
                        if (a[s[i, j] - 1] == 1) { c++; }
                    }
                    //Console.WriteLine();
                    //Console.WriteLine("------");
                    if ((c % 2) != p[i]) { return; }
                }
                count++;
                return;
            }

            //ここでスイッチの状態を固定する処理を書く
            N--;
            a[N] = 0;
            CalcSw(N, a, M, k, s, p);
            a[N] = 1;
            CalcSw(N, a, M, k, s, p);
        }
    }
}