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.io.PrintWriter
val pw = PrintWriter(System.out)
fun main(args: Array<String>) {
    func()
    pw.flush()
}
fun println(value: Any) { pw.println(value) }

fun func() {
    val (N, M) = listOfInt()
    var SW = Array<Int>(N, { 0 })
    for (m in 0 until M) {
        val mm = 1 shl m
        val K = listOfInt().drop(1)
        for (k in K) {
            val sw = k - 1;
            SW[sw] = SW[sw] or mm
        }
    }
    var p = 0
    val P = listOfInt()
    P.forEachIndexed { index, i -> p = p or (i shl index)}

    var ans = 0;
    for (k in 0 until (1 shl N)) {
        var t = 0
        for (n in 0 until N) {
            if (((k ushr n) and 1) != 0) t = t xor SW[n]
        }
        if (t == p) ans++
    }
    println(ans);
}

fun next() = readLine()!!
fun nextInt() = next().toInt()
fun listOfString() = next().split(" ")
fun listOfInt() = listOfString().map { Integer.parseInt(it) } //{ it.toInt() }