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 (
	"bufio"
	"fmt"
	"os"
	"strconv"
)

type light struct {
	k  int
	sl []bool
	p  int
}

func main() {
	sc := bufio.NewScanner(os.Stdin)
	sc.Split(bufio.ScanWords)
	sc.Scan()
	n, _ := strconv.Atoi(sc.Text())
	sc.Scan()
	m, _ := strconv.Atoi(sc.Text())

	l := make([]light, m)
	for i := 0; i < m; i++ {
		sc.Scan()
		k, _ := strconv.Atoi(sc.Text())
		li := light{k, make([]bool, n), i}
		for j := 0; j < k; j++ {
			sc.Scan()
			s, _ := strconv.Atoi(sc.Text())
			li.sl[s-1] = true
		}
		l[i] = li
	}

	for i := 0; i < m; i++ {
		sc.Scan()
		p, _ := strconv.Atoi(sc.Text())
		l[i].p = p
	}

	ans := 0
	for i := 0; i < pow(2, n); i++ {
		sl := convsl(i, n)
		for j, li := range l {
			if check(li.sl, sl)%2 != li.p {
				break
			}
			if j == len(l)-1 {
				ans++
			}
		}
	}
	fmt.Println(ans)
}

func check(x, y []bool) int {
	c := 0
	for i := range x {
		if x[i] && y[i] {
			c++
		}
	}
	return c
}

func convsl(x, n int) []bool {
	r := make([]bool, n)
	for i := 0; i < n; i++ {
		if x%2 == 1 {
			r[i] = true
		}
		x >>= 1
	}
	return r
}

func pow(x, n int) int {
	ans := 1
	for i := 0; i < n; i++ {
		ans *= x
	}
	return ans
}