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 Common Lisp (SBCL 1.1.14) (judge verdict: Accepted)

;; -*- coding: utf-8 -*-
(eval-when (:compile-toplevel :load-toplevel :execute)
  (defparameter OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0)))
  #+swank (ql:quickload '(:cl-debug-print :fiveam))
  #-swank (set-dispatch-macro-character #\# #\> (lambda (s c p) (declare (ignore c p)) (read s nil (values) t))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)
#-swank (disable-debugger) ; for CS Academy

;; BEGIN_INSERTED_CONTENTS
(declaim (ftype (function * (values fixnum &optional)) read-fixnum))
(defun read-fixnum (&optional (in *standard-input*))
  (macrolet ((%read-byte ()
               `(the (unsigned-byte 8)
                     #+swank (char-code (read-char in nil #\Nul))
                     #-swank (sb-impl::ansi-stream-read-byte in nil #.(char-code #\Nul) nil))))
    (let* ((minus nil)
           (result (loop (let ((byte (%read-byte)))
                           (cond ((<= 48 byte 57)
                                  (return (- byte 48)))
                                 ((zerop byte) ; #\Nul
                                  (error "Read EOF or #\Nul."))
                                 ((= byte #.(char-code #\-))
                                  (setf minus t)))))))
      (declare ((integer 0 #.most-positive-fixnum) result))
      (loop
        (let* ((byte (%read-byte)))
          (if (<= 48 byte 57)
              (setq result (+ (- byte 48) (* 10 (the (integer 0 #.(floor most-positive-fixnum 10)) result))))
              (return (if minus (- result) result))))))))

(defmacro dbg (&rest forms)
  #+swank
  (if (= (length forms) 1)
      `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
      `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

(defconstant +mod+ 1000000007)

;; Body

(defun main ()
  (let* ((n (read))
         (as (make-array n :element-type 'uint32))
         (cumuls (make-array (+ n 1) :element-type 'uint62 :initial-element 0))
         (res most-positive-fixnum))
    (declare (uint62 n res))
    (dotimes (i n)
      (setf (aref as i) (read-fixnum)
            (aref cumuls (+ i 1)) (+ (aref cumuls i) (aref as i))))
    (labels ((query (l r)
               (- (aref cumuls r) (aref cumuls l))))
      (loop for pivot from 2 to (- n 2)
            do (labels ((query-pq (i)
                          (abs (- (query 0 i) (query i pivot))))
                        (query-rs (i)
                          (abs (- (query pivot i) (query i n)))))
                 (let* ((qtile1 (sb-int:named-let bisect ((ng 0) (ok pivot))
                                  (declare (uint32 ng ok))
                                  (if (<= (- ok ng) 1)
                                      ok
                                      (let ((mid (ash (+ ok ng) -1)))
                                        (if (>= (query-pq (+ mid 1))
                                                (query-pq mid))
                                            (bisect ng mid)
                                            (bisect mid ok))))))
                        (qtile3 (sb-int:named-let bisect ((ng pivot) (ok n))
                                  (declare (uint32 ng ok))
                                  (if (<= (- ok ng) 1)
                                      ok
                                      (let ((mid (ash (+ ok ng) -1)))
                                        (if (>= (query-rs (+ mid 1))
                                                (query-rs mid))
                                            (bisect ng mid)
                                            (bisect mid ok))))))
                        (p (query 0 qtile1))
                        (q (query qtile1 pivot))
                        (r (query pivot qtile3))
                        (s (query qtile3 n)))
                   (dbg pivot qtile1 qtile3 p q r s)
                   (setq res (min res (- (max p q r s) (min p q r s))))))))
    (println res)))

#-swank (main)