
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
Given a positive integer \( n \) greater than 2004, fill the numbers \( 1, 2, 3, \cdots, n^2 \) into an \( n \times n \) chessboard (composed of \( n \) rows and \( n \) columns) so that each square has exactly one number. A square is called "superior" if the number in that square is greater than the numbers in at least 2004 squares in its row and 2004 squares in its column. Find the maximum number of "superior" squares on the chessboard.

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