Migrated from Lutece 248 To the Max

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Description

Given a two-dimensional array of positive and negative integers, a sub-rectangle is any contiguous sub-array of size $1 \times 1$ or greater located within the whole array. The sum of a rectangle is the sum of all the elements in that rectangle. In this problem the sub-rectangle with the largest sum is referred to as the maximal sub-rectangle.

As an example, the maximal sub-rectangle of the array:

0 -2 -7 0 
9 2 -6 2 
-4 1 -4 1 
-1 8 0 -2

is in the lower left corner:

9 2 
-4 1 
-1 8

and has a sum of $15$.

Input

The input consists of an $N \times N$ array of integers. The input begins with a single positive integer $N$ on a line by itself, indicating the size of the square two-dimensional array. This is followed by $N^2$ integers separated by whitespace (spaces and newlines). These are the $N^2$ integers of the array, presented in row-major order. That is, all numbers in the first row, left to right, then all numbers in the second row, left to right, etc. $N$ may be as large as $100$. The numbers in the array will be in the range $[-127,127]$.

Output

Output the sum of the maximal sub-rectangle.

Samples

4
0 -2 -7 0 9 2 -6 2
-4 1 -4 1 -1
8 0 -2

15

Note

the sub-rectangle shouldn't be empty!!!

The data used in this problem is unofficial data prepared by silentsky. So any mistake here does not imply mistake in the offcial judge data.

Resources

Greater New York 2001