is writing a problem for a programming contest. However, there is one issue: he can't solve it and needs your help.
The problem goes as follows:
Given an ~N \times N~ matrix of numbers, what is the longest strictly increasing path through the matrix, assuming that you can only move horizontally or vertically?
Being the good friend that you are, can you help him do it?
Note: If you are using Python 2/3, submit using PyPy 2/3 instead because it is significantly faster
The first line contains ~N~.
The next ~N~ lines each contain ~N~ space separated integers ~x~ that denote the matrix.
On a line, output the longest strictly increasing path in the matrix.
For all subtasks:
~1 \le x \le 10^9~
Subtask 1 [5%]
~1 \le N \le 2~
Subtask 2 [20%]
~1 \le N \le 50~
Subtask 3 [75%]
~1 \le N \le 1.5 \times 10^3~
Sample Input 1
3 9 8 4 7 2 3 6 1 5
Sample Output 1
To get the longest strictly increasing path you can traverse the values of the matrix in the following order: ~1, 2, 3, 4, 8, 9~. Since path length is calculated by how many movements you have to make and not by the amount of values that you touch, the answer is ~5~ instead of ~6~.
Note that the values aren't guaranteed to be unique in the actual test data.
Sample Input 2
3 1 2 3 8 9 4 7 6 5
Sample Output 2
The longest path is a spiral starting from ~1~ and ending at ~9~.