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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include<string>
#include<iomanip>
#include<iostream>
#include<math.h>
#include<vector>
#include <algorithm>
#include <functional>
using namespace std;
#define NUM 100001
typedef long long ll;

bool desc(pair<string, int > a, pair<string, int > b)
{
	if (a.first != b.first)
		return a.first < b.first;
	else
		return (a.second > b.second);
}

int main() {
	long N, M;
	long k[10];
	long s[10][10];
	long p[10];
	cin >> N >> M;

	long sw[10] = { 0 };

	for (int i = 0; i < M; i++) {
		cin >> k[i];
		for (int j = 0; j < k[i]; j++) {
			cin >> s[i][j];
			s[i][j]--;
		}
	}

	for (int i = 0; i < M; i++) {
		cin >> p[i];
	}

	
	int sum = 0;
	
	// all
	for (int i = 0; i < pow(2,N); i++) {

		// select sw
		for (int j = 0; j < N; j++) {
			sw[j] = (i / int(pow(2, j)) )% 2;
		}


		int boo = 1;
		
		for (int ind = 0; ind < M; ind++) {
			int swsum = 0;
			// swsum
			for (int l = 0; l < k[ind]; l++) {
				if (sw[s[ind][l]] == 1)
					swsum++;
			}
			if (swsum % 2 != p[ind]) {
				boo = 0;
				break;
			}
				
		}

		if (boo == 1)
			sum++;
	}

	cout << sum << endl;

	return 0;
}