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: Accepted)

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

    class Program
    {
        static int n; // switch count
        static int m; // bulb count
        static bulb[] bulbs;

        static void Main(string[] args)
        {
            // read n, m;
            string[] nm = Console.ReadLine().Split(' ');
            n = Convert.ToInt32(nm[0]);
            m = Convert.ToInt32(nm[1]);

            bulbs = new bulb[m];

            for (int i = 0; i < m; i++)
            {
                // read bulb's switch
                bulbs[i] = new bulb()
                {
                    switches = returnSwitch(Console.ReadLine().Split(' '))
                };
            }

            string[] ps = Console.ReadLine().Split(' ');
            for (int i = 0; i < m; i++)
            {
                // read bulb's p
                bulbs[i].p = Convert.ToInt32(ps[i]);
            }

            int count = isBulbsAllOn(true, 0, new bool[n]) + isBulbsAllOn(false, 0, new bool[n]);

            Console.WriteLine(count);
        }

        static int isBulbsAllOn(bool switchOn, int now, bool[] switches)
        {
            // switches option
            switches[now] = switchOn;
            now++;

            if (now < n)
            {
                return isBulbsAllOn(true, now, (bool[])switches.Clone()) + isBulbsAllOn(false, now, switches.ToArray());
            }
            else // all bulbs on check
            {
                for (int i = 0; i < m; i++)
                {
                    bulb b = bulbs[i];
                    int swCnt = 0;
                    for (int j = 0; j < b.switches.Length; j++)
                    {
                        if (switches[b.switches[j]])
                            swCnt++;
                    }
                    if (b.p == 0 && swCnt % 2 == 0) // even
                    {

                    }
                    else if (b.p == 1 && swCnt % 2 == 1) // odd
                    {

                    }
                    else // bulb is off
                        return 0;
                }
                return 1;
            }
        }

        // get switches from string
        static int[] returnSwitch(string[] s)
        {
            int n = Convert.ToInt32(s[0]);
            int[] result = new int[n];
            for (int i = 1; i <= n; i++)
            {
                result[i - 1] = Convert.ToInt32(s[i]) - 1;
            }
            return result;
        }
    }

    class bulb
    {
        public int[] switches;
        public int p = 0;
    }