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(unused_imports)]
use std::collections::HashSet;

#[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 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(|e| e.parse().ok().unwrap())
        .collect()
}


fn main() {
    let nm = read_vec::<usize>();
    let n = nm[0];
    let m = nm[1];

    let mut bitsv : Vec<u32> = Vec::with_capacity(m);
    for _i in 0..m {
        let kss = read_vec::<usize>();
        let k = kss[0];
        let mut bits : u32 = 0;
        for j in 1..k+1 {
            let s = kss[j] as u32;
            bits = bits | (1 << (s-1));
        }
        bitsv.push(bits);
    }

    let ps = read_vec::<u32>();

    let num = solve_bits(m, n, &bitsv, &ps);

    println!("{}", num);
}

fn solve_bits(m: usize, n: usize, bitsv: &[u32], ps: &[u32]) -> u32 {
    let mut total : u32 = 0;
    for x in 0..1<<n {
        let mut tsuku : bool = true;
        for i in 0..m {
            let turned_ons = x & bitsv[i];
            let c_one = count_1(turned_ons);
            if c_one % 2 != ps[i] {
                tsuku = false;
                break;
            }
        }
        if tsuku {
            total += 1;
        }

    }
    return total;
}

fn count_1(x : u32) -> u32 {
    let mut xx = x;
    let mut c : u32 = 0;
    while  xx > 0 {
        if xx & 1 > 0 {
            c += 1;
        }
        xx = xx >> 1;
    }
    return c;
}