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;

#ifndef DEBUG
#define dump(...)
#endif

signed main() {
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);

    size_t N, M;
    cin >> N >> M;

    vector<int> sw[M];
    for (size_t i = 0; i < M; ++i) {
        size_t k; cin >> k;
        for (size_t j = 0; j < k; ++j) {
            int s; cin >> s;
            sw[i].emplace_back(s - 1);
        }
    }

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

    int res = 0;
    for (int bit = 0; bit < (1 << N); ++bit) {
        bool ok = true;
        for (int i = 0; i < M; ++i) {
            int cnt = 0;
            for (auto &si : sw[i]) {
                if (bit & (1 << si)) {
                    cnt++;
                }
            }
            if (cnt % 2 != p[i]) {
                ok = false;
                break;
            }
        }
        if (ok) ++res;
    }

    cout << res << "\n";

    return 0;
}