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)

use proconio::input;
use proconio::marker::Usize1;
use std::collections::HashMap;

fn main() {
  input! {
    n: usize,
    m: usize,
  }
  
  let mut map: HashMap<usize, Vec<usize>> = HashMap::new();
  for i in 0..m {
    input! {
      nn: usize,
      switches: [Usize1;nn]
    }
    map.insert(i, switches.into_iter().collect::<Vec<usize>>());
  }
  
  input! {
    needs: [usize;m]
  }
  
  let mut total = 0;
  for mut i in 0..2usize.pow(n as u32) {
    let mut ons: Vec<usize> = vec![0;n];
    let mut index = 0;
    while 0 <  i {
      ons[index] = i & 1;
      index += 1;
      i >>= 1;
    }
    
    let mut flag = true;
    for key in map.keys() {
      let mut sum = 0;
      for ii in map.get(key).unwrap() {
        sum += ons[*ii];
      }
      if sum % 2 != needs[*key] {
        flag = false;
        break
      }
    }
    
    if flag {
      total += 1;
    }
  }
  println!("{}", total);  
}