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)

import scala.io.{StdIn => in}

object Main extends App {
  val Array(n, m) = in.readLine.split(" ").map(_.toInt)

  val graph = Array.ofDim[Boolean](m,n)
  val ki = Iterator.continually(in.readLine)
    .take(m)
    .map(_.split(" ").map(_.toInt - 1))
    .toArray
  val p = in.readLine.split(" ").map(_.toInt)

  for {
    i <- 0 until m
    j <- 0 until ki(i)(0) + 1
  } {
    graph(i)(ki(i)(j)) = true
  }

  val ans = dfsEnum(2,n)
  println(ans)

  def dfsEnum(d:Int, n:Int):Int = {
    // d進 n個列挙
    val enum = Array.ofDim[Boolean](n)
    var ret = 0
  
    def evalute():Unit = {
      val res = for(i <- 0 until m) yield {
        val cnt = (0 until n)
        .map(j => if(graph(i)(j) && enum(j)) 1 else 0)
        .sum
        cnt % 2 == p(i)
      }
      if(res.forall(_ == true)) ret += 1
    }
  
    def dfs(cnt:Int):Unit = {
      if(cnt == n) evalute
      else {
        for(b <- List(false, true)) {
          enum(cnt) = b
          dfs(cnt+1)
        }
      }
    }
  
    dfs(0)
    ret
  }

}