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.Runtime.InteropServices;
using System.Text;

namespace AtCoder.ABC
{
    class QuestionA
    {
        class KS
        {
            public int K;
            public List<int> S;

            public KS(int[] ks)
            {
                K = ks[0];

                S = new List<int>();

                for(int i = 1; i <= K; i++) {
                    S.Add(ks[i] - 1);
                }
            }
        }

        public static void Main(string[] args)
        {
            var nm = Console.ReadLine().Split(" ").Select(x => int.Parse(x)).ToArray();

            var n = nm[0];
            var m = nm[1];

            var ks = new KS[m];

            for(int i = 0; i < m; i++) {
                ks[i] = new KS(Console.ReadLine().Split(" ").Select(x => int.Parse(x)).ToArray());
            }

            var p = Console.ReadLine().Split(" ").Select(x => int.Parse(x)).ToArray();
            var switches = new bool[n];

            var max = 1 << n;

            var count = 0;

            for(int i = 0; i < max; i++) {
                for(int j = 0; j < n; j++) {
                    switches[j] = ((i >> j) & 1) == 1;
                }

                for(int j = 0; j < m; j++) {
                    var lightOnCount = 0;

                    for(int k = 0; k < ks[j].K; k++) {
                        if(switches[ks[j].S[k]]) {
                            lightOnCount++;
                        }
                    }

                    if(lightOnCount % 2 != p[j]) {
                        break;
                    }

                    if(j == m - 1) {
                        count++;
                    }
                }
            }

            Console.WriteLine(count);
        }
    }
}