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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.util.Scanner;

public class Main {

  static int[][] bulbs;
  static int[] p;

  public static void main(String[] args) {
    Scanner sc = new Scanner(System.in);
    int n = sc.nextInt();
    int m = sc.nextInt();

    bulbs = new int[m][0];
    p = new int[m];

    for (int i = 0; i < m; ++i) {
      int k = sc.nextInt();
      bulbs[i] = new int[k];

      for (int j = 0; j < k; ++j) {
        int s = sc.nextInt();
        bulbs[i][j] = s;
      }
    }

    for (int i = 0; i < m; ++i) {
      p[i] = sc.nextInt();
    }

    System.out.println(combinations(new int[n], n));
  }

  public static int combinations(int[] combination, int switchN) {
    if (switchN > 0) {
      int[] combination1 = combination.clone();
      combination[switchN - 1] = 0;
      combination1[switchN - 1] = 1;
      return combinations(combination, switchN - 1) + combinations(combination1, switchN - 1);
    }
    return checkSwitches(combination) ? 1 : 0;
  }

  public static boolean checkSwitches(int[] combination) {
    boolean res = true;
    for (int j = 0; j < bulbs.length; ++j) {
      int sum = 0;
      for (int i : bulbs[j])
        sum += combination[i - 1];
      res &= sum % 2 == p[j];
    }
    return res;
  }
}