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: Accepted)

import java.util.Scanner

object Main {

  case class Bulb(nSwitch: Int, switches: IndexedSeq[Int], parity: Int)


  def read() = {
    val sc = new Scanner(System.in)
    val nSwitch, nBulb = sc.nextInt()
    val ks = IndexedSeq.fill[(Int, IndexedSeq[Int])](nBulb) {
      val k = sc.nextInt()
      val s = IndexedSeq.fill[Int](k)(sc.nextInt()-1)
      (k, s)
    }
    val p = IndexedSeq.fill(nBulb)(sc.nextInt())
    val bulbs = (ks, p).zipped.map((kse, pe) => Bulb(kse._1, kse._2, pe))
    (nSwitch, nBulb, bulbs)
  }

  def solve(nSwitch: Int, nLight: Int, bulbs: IndexedSeq[Bulb]): Int = {
    //On: 1, Off: 0

    def lights(switchInfo: IndexedSeq[Int], bulb: Bulb): Boolean = {
      bulb.switches.map(switchInfo).sum % 2 == bulb.parity
    }

    val allSwitchInfo = for (switchBinaryInfo <- 0 until (1 << nSwitch)) yield {
      IndexedSeq.tabulate(nSwitch)(k => (switchBinaryInfo >> k) & 1)
    }

    allSwitchInfo.count(switchInfo => bulbs.forall(bulb => lights(switchInfo, bulb)))
  }

  def main(args: Array[String]): Unit = {
    val (nSwitch, nBulbs, bulbs) = read()
    val result = solve(nSwitch, nBulbs, bulbs)
    println(result)
  }
}