D - Equal Cut

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 $P,Q,R,S$.

5
3 2 4 1 2

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 $2$.

10
10 71 84 33 6 47 23 25 52 64

36

7
1 2 3 1000000000 4 5 6

999999994