
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 \( n > 2004 \), fill the numbers \( 1, 2, \ldots, n^2 \) into an \( n \times n \) chessboard (composed of \( n \) rows and \( n \) columns). Each cell contains exactly one number. A cell is called "superior" if the number in the cell is greater than the numbers in at least 2004 other cells in its row and also greater than the numbers in at least 2004 other cells in its column. Find the maximum number of "superior" cells on the chessboard.

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