
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 (consisting of \( n \) rows and \( n \) columns) such that each square contains exactly one number. If a square contains a number that is greater than the numbers in at least 2004 other squares in its row, and also greater than the numbers in at least 2004 other squares in its column, then this square is called an "excellent square." Find the maximum number of "excellent squares" on the chessboard.

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