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)

// #[warn(dead_code)]
// const TOP: u64 = 1000000007;

pub struct Poper<'a> {
	iter: std::str::SplitWhitespace<'a>,
}

impl<'a> Poper<'a> {
	pub fn new(iter: std::str::SplitWhitespace<'a>) -> Poper {
		Poper { iter: iter }
	}
	pub fn read<T: std::str::FromStr>(&mut self) -> T
	where
		<T as std::str::FromStr>::Err: std::fmt::Debug,
	{
		self.iter.next().unwrap().parse().unwrap()
	}
}

fn main() {
	let input = {
		use std::io::Read;
		let mut s = String::new();
		std::io::stdin().read_to_string(&mut s).unwrap();
		s
	};
	let mut iter = Poper::new(input.split_whitespace());

	let n: u64 = iter.read();
	let m: u64 = iter.read();

	use std::collections::HashMap;

	let mut cons: HashMap<u64, Vec<u64>> = HashMap::new();
	let mut vals: HashMap<u64, u32> = HashMap::new();

	for i in 1..(m + 1) {
		let k: u64 = iter.read();
		let mut v: Vec<u64> = vec![];
		for _ in 1..(k + 1) {
			let s: u64 = iter.read();
			v.push(s);
		}
		cons.insert(i, v);
	}
	for i in 1..(m + 1) {
		let p: u32 = iter.read();
		vals.insert(i, p);
	}

	let check = |sw: u64| -> bool {
		for (i, v) in cons.iter() {
			let mut c: u32 = 0;
			for &n in v.iter() {
				let on = sw & (1 << (n - 1));
				if 0 < on {
					c = c ^ 1;
				}
			}
			if let Some(&c2) = vals.get(i) {
				if c2 != c {
					return false;
				}
			}
		}
		true
	};

	let mut cnt: u64 = 0;
	for sw in 0..(1 << n) {
		if check(sw) {
			cnt += 1;
		}
	}

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