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.Linq;
using System.Collections.Generic;

namespace ABC
{
    class ProgramC
    {
        static void Main(string[] args)
        {
            var inputs = Console.ReadLine().Split(" ");
            var n = int.Parse(inputs[0]);
            var m = int.Parse(inputs[1]);

            var list = new List<int>[m];

            for (var i = 0; i < m; i++)
            {
                var inputs1 = Console.ReadLine().Split(" ").Select(x => int.Parse(x) - 1).ToList();
                inputs1.RemoveAt(0);
                list[i] = inputs1;
            }

            var p = Console.ReadLine().Split(" ").Select(x => int.Parse(x)).ToList();

            var result = 0;

            for (var i = 0; i < (1 << n); i++)
            {
                var r = true;

                for (var j = 0; j < m; j++)
                {
                    var switchs = list[j];
                    var onCount = 0;

                    for (var k = 0; k < switchs.Count; k++)
                    {
                        if (((i >> switchs[k]) & 1) == 1)
                        {
                            onCount++;
                        }
                    }

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

                if (r)
                {
                    result++;
                }
            }

            Console.WriteLine(result);
        }
    }
}