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# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Text.RegularExpressions;

namespace AtCoder
{
    class Program
    {
        static void Main(string[] args)
        {
            var line = Console.ReadLine().Split(' ').Select(x => int.Parse(x)).ToArray();
            int N = line[0];
            int M = line[1];

            var switchsEachLightBulb = new List<int>[M];

            for (int i = 0; i < M; i++)
            {
                switchsEachLightBulb[i] = new List<int>();

                var ks = Console.ReadLine().Split(' ').Select(x => int.Parse(x)).ToArray();

                for(int j = 1; j < ks[0] + 1; j++)
                {
                    switchsEachLightBulb[i].Add(ks[j]);
                }
            }

            var p = Console.ReadLine().Split(' ').Select(x => int.Parse(x)).ToArray();

            int ans = 0;

            for (var bits = 0; bits < 1 << N; bits++)
            {
                bool isElectric = true;

                for (int i = 0; i < M; i++)
                {
                    int onCount = 0;

                    for (int j = 0; j < switchsEachLightBulb[i].Count; j++)
                    {
                        if(((bits >> switchsEachLightBulb[i][j] -1) & 1) == 1)
                        {
                            onCount++;
                        }
                    }

                    if(onCount % 2 != p[i])
                    {
                        isElectric = false;
                        break;
                    }
                }

                if (isElectric)
                {
                    ans++;
                }
            }

            Console.WriteLine(ans);
        }
    }
}