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: Runtime Error)

using System;
using static System.Console;
using static System.Math;

public class Hello{
    public static void Main(){
    int honsu = rint();
    int[,] zahyou = inta2(honsu,2);
    bool[,] map = new bool[5001,5001];
    for(int i=0;i<honsu;i++){
        map[zahyou[i,0],zahyou[i,1]] = true;
    }
    int menseki = 0;
    for(int i=0;i<honsu-1;i++){
        for(int j=i+1;j<honsu;j++){
            int sax = zahyou[j,0] - zahyou[i,0];
            int say = zahyou[j,1] - zahyou[i,1];
            
            int[,] check = new int[4,2];
            check[0,0] = zahyou[i,0] - say;
            check[0,1] = zahyou[i,1] + sax;
            check[1,0] = zahyou[j,0] - say;
            check[1,1] = zahyou[j,1] + sax;
            check[2,0] = zahyou[i,0] + say;
            check[2,1] = zahyou[i,1] - sax;
            check[3,0] = zahyou[j,0] + say;
            check[3,1] = zahyou[j,1] - sax;
            if(check[0,0] > 0 &&check[0,1] > 0 &&check[1,0] > 0 &&check[1,1] > 0 ){
            if(map[check[0,0],check[0,1]] == true && map[check[1,0],check[1,1]] == true){
                int tempx = sax*sax;
                int tempy = say*say;
                menseki = Max(menseki,tempx+tempy);
            }
            }
            if(check[2,0] > 0 &&check[2,1] > 0 &&check[3,0] > 0 &&check[3,1] > 0 ){
            if(map[check[2,0],check[2,1]] == true && map[check[3,0],check[3,1]] == true){
                int tempx = sax*sax;
                int tempy = say*say;
                menseki = Max(menseki,tempx+tempy);
            }
            }
            
        }
    }
    WriteLine(menseki);
    
    
    
    
    
    }
    //各種読取
    static string rstr(){
        return ReadLine();
    }
    static int rint(){
        return int.Parse(ReadLine());
    }
    static string[] stra(){
        return ReadLine().Split(' ');
    }
    static int[] inta(){
        string[] read_str_array = ReadLine().Split(' ');
        int[] return_int_array = new int[read_str_array.Length];
        for(int countup_i=0;countup_i<read_str_array.Length;countup_i++){
            return_int_array[countup_i] = int.Parse(read_str_array[countup_i]);
        }
        return return_int_array;
    }
    static int[,] inta2(int num_array,int in_array){
        int[,] int_array2 = new int[num_array,in_array];
        for(int i=0;i<num_array;i++){
            int[] temp_array = inta();
            for(int j=0;j<in_array;j++){
                int_array2[i,j] = temp_array[j];
            }
        }
        return int_array2;
    }
}