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 (1.15.1) (judge verdict: Accepted)

#[allow(dead_code)]
fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

#[allow(dead_code)]
fn readv<T: std::str::FromStr>() -> Vec<T> {
    read::<String>().split_whitespace()
        .map(|e| e.parse().ok().unwrap()).collect()
}

#[allow(dead_code)]
fn readvv<T: std::str::FromStr>(n: usize) -> Vec<Vec<T>> {
    (0..n).map(|_| readv()).collect()
}

fn main() {
    let (n, m) = {
        let tv = readv();
        (tv[0], tv[1])
    };
    // println!("n: {}, m: {}", n, m);

    let mut ss = vec![];
    for _ in 0..m {
        let s = readv::<usize>();
        let s = s.into_iter().skip(1).map(|i| i - 1).collect::<Vec<usize>>();
        ss.push(s);
    }

    // println!("ss: ");
    // for s in ss {
    //     println!("{:?}", s);
    // }
    // println!("----");

    let pp = readv::<usize>();

    // println!("{:?}", pp);

    let pf = |ls, i| {
        let mut count = 0;
        for sv in ls {
            // println!("        sv: {}, i: {}", sv, i);
            if i & (0x1 << sv) > 0 {
                // println!("          OKOK");
                count += 1;
            }
        }
        count % 2
    };
    // zentansaku
    let mut count = 0;
    for i in 0..(1 << n) {
        let ok = ss.iter().enumerate().all(|(l, s)| {
            // println!("i: {} , l: {}, s: {:?}", i, l, s);
            let pv = pf(s, i);
            let p = pp[l];
            // println!("  (pv: {}, p: {})", pv, p);
            pv == p
        });
        if ok {
            count += 1;
        }
    }
    println!("{}", count);
}