
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 from 1 to \( n^2 \) into the squares of an \( n \times n \) chessboard (which consists of \( n \) rows and \( n \) columns) such that each square has exactly one number. A square is called a "superior square" if the number filled in this square is greater than the numbers filled in at least 2004 squares of its row and at least 2004 squares of 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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