Given a cube of positive and negative integers, find the sub-cube with the largest sum. The sum of a cube is the sum of all the elements in that cube. In this problem, the sub-cube with the largest sum is referred to as the maximal sub-cube. A sub-cube is any contiguous sub-array of size 1x1x1 or greater located within the whole array. As an example, if a cube is formed by following 3x3x3 integers: 0 -1 3 -5 7 4 -8 9 1 -1 -3 -1 2 -1 5 0 -1 3 3 1 -1 1 3 2 1 -2 1 Then its maximal sub-cube which has sum 31 is as follows: 7 4 9 1 -1 5 -1 3 3 2 -2 1

Each input set consists of two parts. The first line of the input set is a single positive integer N between 1 and 20, followed by NxNxN integers separated by white-spaces (newlines or spaces). These integers make up the array in a plane, row-major order (i.e., all numbers on the first plane, first row, left-to-right, then the first plane, second row, left-to-right, etc.). The numbers in the array will be in the range [-127,127]. The input is terminated by a value 0 for N.

The output is the sum of the maximal sub-cube.

Asia 1997, Shanghai (Mainland China)