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;

class Program
{
    static void Main(string[] args)
    {
        var inputs = Console.ReadLine().Split(" ");

        // N個のスイッチとM個の電球
        var N = int.Parse(inputs[0]);
        var M = int.Parse(inputs[1]);

        // i番目の電球はk個のスイッチに繋がっており、スイッチsのうちonになっているスイッチの個数%2がp[i]に等しい時に点灯する
        var k = new List<int>();
        var s = new List<List<int>>();
        var p = new List<int>();

        for(int i = 0;i < M; i++)
        {
            var switchs = Console.ReadLine().Split(" ");
            k.Add(int.Parse(switchs[0]));
            s.Add(new List<int>());
            for (int j = 1; j <= k[i]; j++)
            {
                s[i].Add(int.Parse(switchs[j]));
            } 
        }

        var ps = Console.ReadLine().Split(" ");
        for(int i = 0; i < M; i++)
        {
            p.Add(int.Parse(ps[i]));
        }

        //　電球の点灯パターンを網羅する
        var switches = new bool[N];
        var numOfPatterns = 0;
        for (int bit = 0; bit < (1 << N); bit++)
        {
            for (int i = 0; i < N; ++i)
            {
                if ((bit & (1 << i)) != 0)
                {
                    switches[i] = true;
                }
                else
                {
                    switches[i] = false;
                }
            }

            // 各電球ごとに判定
            var result = true;
            for (int i = 0; i < M; i++)
            {
                var count = 0;
                for(int j = 0; j < k[i]; j++)
                {
                    if (switches[s[i][j] - 1]) count++;
                }
                
                // 点灯するか判定、点灯しなかったら後ろの電球はチェック不要
                if(count % 2 != p[i])
                {
                    result = false;
                    break;
                } 
            }

            if (result) numOfPatterns++;
        }
        Console.WriteLine(numOfPatterns);
    }
}