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 Rust (1.42.0) (judge verdict: Wrong Answer)

fn solve(ksss: Vec<Vec<u64>>, ps: Vec<u64>, n: usize) -> usize {
    let mut result = 0;
    let sss: Vec<Vec<u64>> = ksss.iter()
        .map(|kss| kss.iter().skip(1).map(|s| s - 1).collect::<Vec<_>>())
        .collect();

    for bins in binary_combinations(n) {
        if (0..ps.len()).all(|i| sss[i].iter().zip(&bins).filter(|(_, &bin)| bin).count() % 2 == ps[i] as usize) {
            result += 1
        }
    }

    result
}

fn binary_combinations(n: usize) -> Vec<Vec<bool>> {
    fn rec(n: usize, mut acc: Vec<Vec<bool>>) -> Vec<Vec<bool>> {
        if n == 0 {
            acc
        } else {
            let mut tmp = acc.clone();
            acc.iter_mut().for_each(|v| v.push(false));
            tmp.iter_mut().for_each(|v| v.push(true));
            acc.extend(tmp);

            rec(n - 1, acc)
        }
    }

    rec(n, vec![vec![]])
}

fn main() {
    let (n, m) = read_tup2();
    let ksss = read_vec_n(m);
    let ps = read_vec();

    println!("{}", solve(ksss, ps, n));
}

fn read_tup2<T: std::str::FromStr + Copy>() -> (T, T) {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    let v = s.trim().split_whitespace()
        .map(|c| c.parse().ok().unwrap())
        .collect::<Vec<T>>();
    (v[0], v[1])
}

fn read_vec_n<T: std::str::FromStr>(n: usize) -> Vec<Vec<T>> {
    let mut ss = String::new();
    for _ in 0..n {
        std::io::stdin().read_line(&mut ss).ok();
    }
    let mut v = ss.split("\n").collect::<Vec<&str>>();
    v.pop();
    v.iter()
        .map(
            |s| {
                s.trim().split_whitespace()
                    .map(|c| c.parse().ok().unwrap())
                    .collect()
            }
        )
        .collect()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().split_whitespace()
        .map(|c| c.parse().ok().unwrap())
        .collect()
}