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 <iostream>
#include <string>
#include <vector>
#include <algorithm>
#include <map>
#include <set>
#include <cmath>
#include <random>
#include <queue>

using namespace std;


class game{
public:
	int N, M;
	vector< vector<int> > Ss;
	vector<int> Ps;
	int ans = 0;

	void Input(){
		cin >> N >> M;
		int k;
		int s;
		for(int mi = 0; mi < M; mi++){
			cin >> k;
			vector<int> ls;
			for(int ki = 0 ; ki < k; ki++){
				cin >> s;
				ls.push_back(s-1);
			}
			Ss.push_back(ls);
		}

		int p;
		for(int ni = 0; ni < M; ni++){
			cin >> p;
			Ps.push_back(p);
		}
	}

	int TryS(int ni, vector<int> ls)
	{
		if (ni == N){
			int si;
			int sum;
			int flag = 1;
			for (int mi = 0; mi < M; mi++){
				sum = 0;
				for (int ssi = 0; ssi < Ss[mi].size(); ssi++){
					si = Ss[mi][ssi];
					sum += ls[si];
				}
				if (sum % 2 != Ps[mi]){
					flag = 0;
					break;
				}
			}
			if (flag) {
				ans++;
			}
			return 0;
		}

		vector<int> new_ls = ls;
		new_ls[ni] = 0;
		TryS(ni+1, new_ls);
		new_ls[ni] = 1;
		TryS(ni+1, new_ls);

		return 1;
	}

};


int main()
{

	game g;

	g.Input();
	g.TryS(0, vector<int>(g.N,0));

	printf("%d",g.ans);


	return 0;
}