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: Accepted)

// -*- coding:utf-8-unix -*-

use proconio::derive_readable;
use proconio::fastout;
use proconio::input;
use std::convert::TryInto;
use std::io::*;
use std::str::FromStr;

pub fn read<T: FromStr>() -> T {
    let stdin = stdin();
    let stdin = stdin.lock();
    let token: String = stdin
        .bytes()
        .map(|c| c.expect("failed to read char") as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().expect("failed to parse token")
}

//abc128-C

fn main() {
    //     input! {
    //         n: u8,
    //         m: u8,
    //         sk: [[u8]; n],
    //         ps: [u8; m]
    //     }

    //     println!(
    //         r#"
    // n: {},
    // m: {},
    // sk: {:?},
    // ps: {:?}
    //         "#,
    //         n, m, sk, ps
    //     );

    let n: usize = read();
    let m: usize = read();
    let mut s: Vec<Vec<usize>> = vec![vec![]; m];
    let mut p: Vec<usize> = vec![];

    for i in 0..m {
        let k = read();
        for _ in 0..k {
            s[i].push(read());
        }
    }

    for _ in 0..m {
        p.push(read());
    }
    let mut ans = 0;

    let mut sstate = vec![false; n];

    for switch_state in 0..(2u64.pow(n.try_into().unwrap())) {
        for i in 0..n {
            sstate[i] = (switch_state >> i) % 2 == 0
        }

        let mut is_ok = true;
        for i in 0..m {
            let mut sum = 0;
            for s in &s[i] {
                sum += if sstate[s - 1] { 1 } else { 0 }
            }

            if sum % 2 != p[i] {
                is_ok = false;
            }
        }

        if is_ok {
            ans += 1;
        }
    }

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