Problem:
Score : 600 points

Problem Statement

Snuke has an integer sequence A of length N.

He will make three cuts in A and divide it into four (non-empty) contiguous subsequences B, C, D and E.
The positions of the cuts can be freely chosen.

Let P,Q,R,S be the sums of the elements in B,C,D,E, respectively.
Snuke is happier when the absolute difference of the maximum and the minimum among P,Q,R,S is smaller.
Find the minimum possible absolute difference of the maximum and the minimum among P,Q,R,S.

Constraints

4 \leq N \leq 2 \times 10^5

1 \leq A_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Find the minimum possible absolute difference of the maximum and the minimum among P,Q,R,S.

Sample Input 1

5
3 2 4 1 2

Sample Output 1

2

If we divide A as B,C,D,E=(3),(2),(4),(1,2), then P=3,Q=2,R=4,S=1+2=3.
Here, the maximum and the minimum among P,Q,R,S are 4 and 2, with the absolute difference of 2.
We cannot make the absolute difference of the maximum and the minimum less than 2, so the answer is 2.

Sample Input 2

10
10 71 84 33 6 47 23 25 52 64

Sample Output 2

36

Sample Input 3

7
1 2 3 1000000000 4 5 6

Sample Output 3

999999994

Solution in Kotlin (1.0.0) (judge verdict: Accepted)

fun main(args: Array<String>) {
  val N = readInt()
  val A = readLongs().toMutableList()
  val cumSumA = Array(N) { 0L }
  cumSumA[0] = A[0]
  for (i in 1 until N) {
    cumSumA[i] = cumSumA[i - 1] + A[i]
  }

  var min = Long.MAX_VALUE

  var iL = 0
  var iR = 2
  for (iM in 1 until N - 2) {
    val sumL = cumSumA[iM]
    val sumR = cumSumA[N - 1] - sumL

    while (true) {
      val l = cumSumA[iL]
      val r = sumL - l
      val ll = cumSumA[iL + 1]
      val rr = sumL - ll
      if (Math.abs(l - r) <= Math.abs(ll - rr)) break
      iL++
    }
    while (true) {
      val l = cumSumA[iR] - sumL
      val r = sumR - l
      val ll = cumSumA[iR + 1] - sumL
      val rr = sumR - ll
      if (Math.abs(l - r) <= Math.abs(ll - rr)) break
      iR++
    }

    val d0 = cumSumA[iL]
    val d1 = sumL - d0
    val d2 = cumSumA[iR] - sumL
    val d3 = sumR - d2
    val dList = listOf(d0, d1, d2, d3)

    min = Math.min(min, dList.max()!! - dList.min()!!)
  }
  println(min)
}

fun gcd(a: Int, b: Int): Int = if (a < b) gcd(b, a) else if (b == 0) a else gcd(b, a % b)
fun lcm(a: Int, b: Int): Int = (a * b) / gcd(a, b)

fun readString() = readLine()!!
fun readInt() = readLine()!!.toInt()
fun readInts() = readLine()!!.split(" ").map { it.toInt() }
fun readLongs() = readLine()!!.split(" ").map { it.toLong() }