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: Wrong Answer)

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

namespace competitive_programming
{
    class Program
    {
        static void Main(string[] args)
        {
            int[] input = Console.ReadLine().Split(' ').Select(value => int.Parse(value)).ToArray();
            int n = input[0];
            int m = input[1];

            string[] swinfo = new string[m];
            for(int i = 0;i < m; i++)
            {
                swinfo[i] = Console.ReadLine();
            }

            int[] ps = Console.ReadLine().Split(' ').Select(value => int.Parse(value)).ToArray();

            var lamps = new lamp[m];
            for(int i = 0;i < m;i++)
            {
                var info = swinfo[i].Split(' ').Select(value => int.Parse(value)).ToArray();
                var sw = new int[info.Count() - 1];
                for (int j = 0; j < sw.Length; j++) sw[j] = info[j + 1];
                //Console.WriteLine("sw" + sw.Length);
                lamps[i] = new lamp( ps[i],sw );
            }

            check(1, lamps, n);
            Console.WriteLine(count);

        }

        static int count = 0;

        public static bool check(int swNum,lamp[] lamps,int n)
        {
            Console.WriteLine(swNum);
            if (swNum == n + 1) 
            { 
                foreach(var lamp in lamps)
                {
                    Console.WriteLine(lamp.check);
                    if (lamp.check) continue;
                    else return false;
                }
                count++;
                Console.WriteLine("goal");
                return true;
            }


            if (check(swNum + 1, lamps, n))
            {
                return true;
            }


            var nlamps = lamps;
            for(int i = 0;i < nlamps.Length;i++)
            {
                Console.WriteLine(i + "番目の最初は" + nlamps[i].switches[0]); ;
                if (nlamps[i].switches.Contains(swNum))
                {
                    Console.WriteLine(i +" contain " + nlamps[i].check);
                    nlamps[i].check = !nlamps[i].check;
                }
                
            }

            if (check(swNum + 1, nlamps, n))
            {
                return true;
            }

            return false;
        }
    }


    struct lamp
    {
        public lamp(int p,int[] switches)
        {
            
            this.p = p;
            this.switches = switches;

            if (p == 0) check = true;
            else check = false;

        }

        public bool check;
        public int p;
        public int[] switches;
    }

    struct Vector2
    {
        public Vector2(int x,int y)
        {
            this.x = x;
            this.y = y;
        }
        public int x;
        public int y;
    }

    struct Vector3
    {
        public int x;
        public int y;
        public int z;
    }
}