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)

fun main(args : Array<String>) {
    val (N ,M) = readLine()!!.split(" ").map { it.toInt() }
    val S = mutableListOf<List<Int>>()
    (1..M).forEach {
        val kAndS = readLine()!!.split(" ").map { it.toInt() }
        S.add(kAndS.subList(1, kAndS.size))
    }
    val P = readLine()!!.split(" ").map { it.toInt() }

    val allConbinations = Math.pow(2.toDouble(), N.toDouble()).toInt()

    var answer = 0
    // O(2^N)
    for (bits in 1..allConbinations) {
        // O(N)
        val switchFlags = mutableListOf<Boolean>()
        for (i in 0..(N - 1)) {
            val iThSwitchOn = isBitSet(bits, i)
            switchFlags.add(iThSwitchOn)
        }

        var allFlashed = true
        // o(M)
        S.forEachIndexed { index, s ->
            if (allFlashed) {
                val p = P[index]
                var onCount = 0
                // O(K)
                s.forEach { switchIndex ->
                    if (switchFlags[switchIndex - 1]) {
                        onCount++
                    }
                }
                if (onCount.mod(2) != p) {
                    allFlashed = false
                }
            }
        }

        if (allFlashed) {
            answer++
        }
    }

    println(answer)
}
fun isBitSet(bit: Int, index: Int) : Boolean {
    return bit.shr(index).and(1) == 1
}