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 {
  def main(args: Array[String]) {
    if (args.length > 0 && args(0) == "TEST") {
      println(test())
    } else {
      val input = io.Source.stdin.getLines().mkString("\n")
      println(solve(input).trim())
    }
  }

  def numWithNbits(n: Int): Int = (1 << n) - 1
  def checkBitOn(n: Int, b: Int): Int = (n >> (b - 1)) & 1
  def countBitsOn(n: Int, bl: List[Int]): Int = bl match {
    case Nil => 0
    case bh::bt => checkBitOn(n, bh) + countBitsOn(n, bt)
  }

  def isAllOn(n: Int, is: List[(List[Int], Int)]): Boolean = is match {
    case Nil => true
    case (bits, parity) :: irs => {
      val bitsCount = countBitsOn(n, bits)
      if (bitsCount % 2 != parity) false
      else isAllOn(n, irs)
    }
  }

  def solve(input: String): String = {
    val s: Scanner = new Scanner(input)
    val (n, m) = (s.nextInt, s.nextInt)
    val ks = List.fill(m)(List.fill(s.nextInt)(s.nextInt))
    val ps = List.fill(m)(s.nextInt)
    val is = ks.zip(ps)
    (0 to numWithNbits(n)).count(i => isAllOn(i, is)).toString()
  }

  val tests = List(
    """2 2
      |2 1 2
      |1 2
      |0 1
    """.stripMargin -> """1""",
    """2 3
      |2 1 2
      |1 1
      |1 2
      |0 0 1
    """.stripMargin -> """0""",
    """5 2
      |3 1 2 5
      |2 2 3
      |1 0
    """.stripMargin -> "8"
  )

  def test(): String = tests.map { case (i, o) => (i.trim(), o.trim()) }
    .zipWithIndex.map {
    case ((input, outputExpect), i) =>
      val output = solve(input).trim()
      val tnum = i + 1
      s"test$tnum:" + (
        if (output == outputExpect) "Passed"
        else s"Failed\nexpect:\n$outputExpect\noutput:\n$output")
  }
    .mkString("\n")
}