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 (2.13.1) (judge verdict: Accepted)

object Main {

  def main(args: Array[String]): Unit = {
    val sc = new java.util.Scanner(System.in)

    val N, M = sc.nextInt()
    val S = Array.fill(M)(0)
    for {
      i <- 0 until M
    } {
      val ki = sc.nextInt()

      for {
        _ <- 0 until ki
      } {
        val s = sc.nextInt()
        S(i) |= (1 << (s - 1))
      }
    }
    val P = Array.fill(M)(sc.nextInt())

    // 電球iのスイッチは, マスクを与えている.
    // (1 << sij)のfoldでマスクを形成し, ビットと&をとって, ビットを足し合わせて2で割った場合, piに等しければ点灯する
    // つまり, 全てのスイッチの状態を変化させ
    // 全ての電球について, その状態で, 全て条件を満たす場合につき, カウントする
    val r = (for {
      s <- 0 until Math.pow(2, N).toInt
      if (0 until M).forall { k =>
        (0 until N).foldLeft(0)((sum, b) => if (((s & (1 << b)) & S(k)) == (1 << b)) (sum + 1) % 2 else sum) == P(k)
      }
    } yield s).size

    println(r)
  }

}