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)

//---------- begin scannner ----------
mod scanner {
    use std::str::FromStr;
    use std::str::SplitWhitespace;
    use std::io::Read;
    use std;
    pub struct Scanner<'a> {
        it: SplitWhitespace<'a>
    }
    impl<'a> Scanner<'a> {
        pub fn new(s: &'a String) -> Scanner<'a> {
            Scanner {
                it: s.split_whitespace()
            }
        }
        pub fn next<T: FromStr>(&mut self) -> T {
            match self.it.next().unwrap().parse::<T>() {
                Ok(v) => v,
                _ => panic!("Scanner error")
            }
        }
    }
    pub fn read_string() -> String {
        let mut s = String::new();
        std::io::stdin().read_to_string(&mut s).unwrap();
        s
    }
}

//---------- end scannner ----------

fn run() {
    let sc = scanner::read_string();
    let mut sc = scanner::Scanner::new(&sc);
    let n: usize = sc.next();
    let m: usize = sc.next();
    let mut s = vec![vec![]; m];
    for i in 0..m {
        let k: usize = sc.next();
        for _ in 0..k {
            s[i].push(sc.next::<u32>() - 1);
        }
    }
    let mut p: Vec<usize> = vec![];
    for _ in 0..m {
        p.push(sc.next());
    }
    let mut ans = 0;
    for i in 0..(1 << n) {
        let mut ok = 1;
        for j in 0..m {
            let mut c = 0;
            for &k in &s[j] {
                c ^= (i >> k) & 1;
            }
            if c != p[j] {
                ok = 0;
            }
        }
        ans += ok;
    }
    println!("{}", ans);
}

fn main() {
    run();
}