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 <bits/stdc++.h>
using namespace std;
typedef long long ll;
#define rep(i, begin, end) for (int i = begin; i < (int)(end); i++)
#define all(v) v.begin(), v.end()
const int MOD = 1000000007;

int main()
{
    int N, M;
    cin >> N >> M;
    vector<vector<int>> s(M, vector<int>(10, -1));
    vector<int> p(M), count(M, 0);
    rep(i, 0, M)
    {
        int k;
        cin >> k;
        rep(j, 0, k)
        {
            cin >> s.at(i).at(j);
        }
    }
    rep(i, 0, M)
    {
        cin >> p.at(i);
    }
    ll ans = 0;
    bool f = true;
    for (int tmp = 0; tmp < (1 << N); tmp++)
    {
        bitset<10> bit(tmp);
        rep(i, 0, M)
        {
            rep(j, 0, 10)
            {
                if (s.at(i).at(j) != -1)
                {
                    if (bit.test(s.at(i).at(j) - 1))
                    {
                        count.at(i)++;
                    }
                }
            }
            if (count.at(i) % 2 != p.at(i))
            {
                f = false;
            }
            count.at(i) = 0;
        }
        if (f)
        {
            ans++;
        }
        else
        {
            f = true;
        }
    }
    cout << ans << endl;
}