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 kotlin.comparisons.compareBy

fun main(args: Array<String>) {
    val lines: MutableList<List<Int>> = mutableListOf()
    while (true) {
        val str = readLine()
        if (str != null) {
            lines.add(str.split(" ").map { Integer.parseInt(it) })
        } else {
            break
        }
    }
    val n = lines[0][0]
    val m = lines[0][1]

    var switchList = (1..n).map { Switch() }
    val pp = lines.last()

    val lightList: MutableList<Light> = mutableListOf()
    (1..m).map {
        val light = Light(oe = pp[it-1])
        val line = lines[it]
        val k = line[0]
        (1..k).map { switchList[line[it]-1] }
                .forEach { it.connected.add(light) }
        lightList.add(light)
    }

    switchList = switchList.sortedWith(compareBy { it.connected.size })

    val case = 2 shl (n - 1)

    var count = 0
    (0 until case).forEach { cn ->
//        println("case: " + cn)
        (0 until n).forEach {
            switchList[it].switchTo((cn shr it) and 1)
        }
        if (lightList.all { it.on() }) count++
    }
    print(count)
}

data class Light(val oe: Int, var count: Int = 0) {
    fun on(): Boolean {
//        println(this)
        return count % 2 == oe
    }
}
data class Switch(val connected: MutableList<Light> = emptyList<Light>().toMutableList(), var state: Int = 0) {
    fun switchTo(state: Int) {
        if (state != this.state) {
            this.state = state
            this.connected.forEach {
                if (state == 0) {
                    it.count--
                } else {
                    it.count++
                }
            }
        }
    }
}