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 Scala (2.11.7) (judge verdict: Wrong Answer)

object Main {

  import Math.pow

  def main(args: Array[String]): Unit = {
    val sc = new java.util.Scanner(System.in)
    val N = sc.nextInt
    val M = sc.nextInt
    val k = new Array[Int](M)
    val s = new Array[Long](M)
    for (i <- 0 until M) {
      k(i) = sc.nextInt
      var tmp = 0L
      for (j <- 0 until k(i)) {
        tmp += pow(10.0, sc.nextInt - 1).toLong
      }
      s(i) = tmp
    }

    val p = Array.fill(M)(sc.nextInt)
    var result = 0L

    def countOn(b: Int): Unit = {
      for (i <- 0 until M) {
        val on = b & s(i)
        if (on.toBinaryString.count(_.==('1')) % 2 != p(i)) {
          return
        }
      }
      result += 1
    }

    for (b <- 0 until pow(2.0, N).toInt) {
      countOn(b)
    }
    println(result)
  }
}