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)

use std::str::FromStr;
use std::io::Read;
use std::ops::Sub;
use std::convert::From;

pub trait Convert {
    fn to_val<T>(&mut self) -> T
    where
        T: FromStr;

    fn to_vec<T>(&mut self, n: usize) -> Vec<T>
    where
        T: FromStr;

    fn to_vec_1origin<T>(&mut self, n: usize) -> Vec<T>
    where
        T: FromStr + Sub<T, Output = T> + From<u8>;

    fn to_vecs_1origin<T>(&mut self, n: usize) -> Vec<Vec<T>>
    where
        T: FromStr + Sub<T, Output = T> + From<u8>;
}

impl<'a> Convert for ::std::str::SplitWhitespace<'a> {
    fn to_val<T>(&mut self) -> T
    where
        T: FromStr,
    {
        let word = self.next().unwrap();
        word.parse().ok().unwrap()
    }

    fn to_vec<T>(&mut self, n: usize) -> Vec<T>
    where
        T: FromStr,
    {
        (0..n)
            .map(|_| self.next().unwrap().parse().ok().unwrap())
            .collect::<Vec<_>>()
    }

    fn to_vec_1origin<T>(&mut self, n: usize) -> Vec<T>
    where
        T: FromStr + Sub<T, Output = T> + From<u8>,
    {
        (0..n)
            .map(|_| {
                self.next().unwrap().parse::<T>().ok().unwrap() - T::from(1)
            })
            .collect::<Vec<_>>()
    }

    fn to_vecs_1origin<T>(&mut self, n: usize) -> Vec<Vec<T>>
    where
        T: FromStr + Sub<T, Output = T> + From<u8>,
    {
        (0..n)
            .map(|_| {
                let val = self.to_val();
                self.to_vec_1origin(val)
            })
            .collect::<Vec<_>>()
    }
}

pub fn read_stdin() -> String {
    let mut s = String::new();
    ::std::io::stdin().read_to_string(&mut s).unwrap();
    s
}

type ConnectedList = Vec<Vec<usize>>;
fn evaluate(v: &ConnectedList, ps: &Vec<u64>, on: Vec<bool>) -> bool {
    for (ss, &p) in v.iter().zip(ps.iter()) {
        let c = ss.iter().filter(|&s| on[*s]).count() as u64;
        if c % 2 != p {
            return false;
        }
    }
    return true;
}

fn force(v: &ConnectedList, ps: &Vec<u64>, n: usize) -> u64 {
    (0..2_u32.pow(n as u32))
        .filter(|on_src| {
            let on: Vec<bool> = (0..n).map(|i| ((on_src >> i) & 1) == 1).collect();
            evaluate(v, ps, on)
        })
        .count() as u64
}

fn main() {
    let source = read_stdin();
    let mut iter = source.split_whitespace();
    let n: usize = iter.to_val();
    let m: usize = iter.to_val();
    let v: ConnectedList = iter.to_vecs_1origin(m);
    let ps: Vec<u64> = iter.to_vec(m);

    println!("{}", force(&v, &ps, n));
}