
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let  $n$  be a positive integer. An  $n\times n$  board consisting of  $n^2$  cells, each being a unit square colored either black or white, is called *convex* if for every black colored cell, both the cell directly to the left of it and the cell directly above it are also colored black. We define the *beauty* of a board as the number of pairs of its cells  $(u,v)$  such that  $u$  is black,  $v$  is white, and  $u$  and  $v$  are in the same row or column. Determine the maximum possible beauty of a convex  $n\times n$  board.

*Proposed by Ivan Novak*

Present the answer in LaTex format: \boxed{Your answer}
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