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 Kotlin (1.0.0) (judge verdict: Accepted)

import java.util.BitSet

fun main(args: Array<String>) {
    val (n, m) = readInputLine().split(" ").map{it.toInt()}
    
    val switches = mutableListOf<List<Int>>()
    
    for (i in 1..m) {
        val s = readInputLine().split(" ").map{it.toInt() - 1}
        switches.add(s.drop(1))
    }
    
    val p = readInputLine().split(" ").map{it.toInt()}

    println(dfs(0, n, BitSet(n), p, switches))
}

fun readInputLine(): String {
    return readLine()!!
}

fun dfs(index:Int, n: Int, s: BitSet, p: List<Int>, switches: List<List<Int>>): Int {
    if (index == n) {
        var ok = true
        for ((i, switch) in switches.withIndex()) {
            var on = 0
            for (k in switch) {
                if (s[k]) {
                    on++
                }
            }
            if (on % 2 != p[i]) {
                ok = false
                break
            }
        }
        if (ok) {
            return 1
        } else {
            return 0
        }
    }
    
    var ret = 0
    
    ret += dfs(index + 1, n, s, p, switches)
    
    val newSet = s.clone() as BitSet
    newSet[index] = true
    ret += dfs(index + 1, n, newSet, p, switches)

    return ret
}