Problem:
Score : 300 points

Problem Statement

The season for Snuke Festival has come again this year. First of all, Ringo will perform a ritual to summon Snuke. For the ritual, he needs an altar, which consists of three parts, one in each of the three categories: upper, middle and lower.

He has N parts for each of the three categories. The size of the i-th upper part is A_i, the size of the i-th middle part is B_i, and the size of the i-th lower part is C_i.

To build an altar, the size of the middle part must be strictly greater than that of the upper part, and the size of the lower part must be strictly greater than that of the middle part. On the other hand, any three parts that satisfy these conditions can be combined to form an altar.

How many different altars can Ringo build? Here, two altars are considered different when at least one of the three parts used is different.

Constraints

1 \leq N \leq 10^5

1 \leq A_i \leq 10^9(1\leq i\leq N)

1 \leq B_i \leq 10^9(1\leq i\leq N)

1 \leq C_i \leq 10^9(1\leq i\leq N)

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 ... A_N
B_1 ... B_N
C_1 ... C_N

Output

Print the number of different altars that Ringo can build.

Sample Input 1

2
1 5
2 4
3 6

Sample Output 1

3

The following three altars can be built:

Upper: 1-st part, Middle: 1-st part, Lower: 1-st part

Upper: 1-st part, Middle: 1-st part, Lower: 2-nd part

Upper: 1-st part, Middle: 2-nd part, Lower: 2-nd part

Sample Input 2

3
1 1 1
2 2 2
3 3 3

Sample Output 2

27

Sample Input 3

6
3 14 159 2 6 53
58 9 79 323 84 6
2643 383 2 79 50 288

Sample Output 3

87

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 = sc.nextInt()
    val A = Array.fill(N)(sc.nextInt()).sorted
    val B = Array.fill(N)(sc.nextInt())
    val C = Array.fill(N)(sc.nextInt()).sorted(Ordering.Int.reverse)

    // 3カテゴリーのパーツがそれぞれN個ずつあります
    // パーツのサイズ
    // 中部のパーツのサイズは、上部のパーツのサイズより真に大きく
    // 下部のパーツのサイズは、中部のパーツのサイズより真に大きくなければなりません
    // 作れる祭壇は何種類あるでしょうか/
    // ただし 2つの祭壇が異なるとは, 上部, 中部, 下部に使われるピースのうち少なくとも1つが異なる
    // 1つ決めた時の, そのパーツより大きいパーツはO(log N)で出すことができる
    // 普通にやると, O(N^2logN)とかになる, 間に合わない
    // なので, 中部を先に決めておく
    // 中部を決めた時, その時上部のパーツに使えるのは, 中部より小さいもの
    // 下部のパーツに使えるのは, 中部より大きいもの,
    // これらは それぞれ O(log N)で求めることができ, O(NlogN)で求めることができる

    def upperBounds(A: Array[Int], b: Int): Int = {
      def recursive(l: Int, r: Int): Int =
        if (r - l > 1) {
          val mid = l + (r - l) / 2
          if (A(mid) >= b) recursive(l, mid)
          else recursive(mid, r)
        } else l
      recursive(-1, N)
    }

    def lowerBounds(C: Array[Int], b: Int): Int = {
      def recursive(l: Int, r: Int): Int =
        if (r - l > 1) {
          val mid = l + (r - l) / 2
          if (C(mid) <= b) recursive(l, mid)
          else recursive(mid, r)
        } else l
      recursive(-1, N)
    }

    val r = (0 until N).foldLeft(0L) { case (sum, i) =>
      sum + (upperBounds(A, B(i)) + 1L) * (lowerBounds(C, B(i)) + 1L)
    }

    println(r)
  }

}