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 Go (1.6) (judge verdict: Accepted)

package main

import "fmt"

var (
	xs []int
	s  [][]int
	p  []int
)

func dfs(cur int, sw int) int {
	if cur != 0 {
		xs[cur-1] = sw
	}
	if cur == len(xs) {
		for i := range s {
			var light int
			for j := range s[i] {
				light += xs[s[i][j]-1]
			}
			if light%2 != p[i] {
				return 0
			}
		}
		return 1
	}
	sum := 0
	sum += dfs(cur+1, 0)
	sum += dfs(cur+1, 1)
	return sum
}

func main() {
	var n, m int
	fmt.Scan(&n, &m)
	s = make([][]int, m)
	var k int
	for i := range s {
		fmt.Scan(&k)
		s[i] = make([]int, k)
		for j := 0; j < k; j++ {
			fmt.Scan(&s[i][j])
		}
	}
	p = make([]int, m)
	for i := range p {
		fmt.Scan(&p[i])
	}

	xs = make([]int, n)
	fmt.Println(dfs(0, 0))
}