
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 fixed positive integer and consider an  $n\times n$  grid of real numbers. Determine the greatest possible number of cells  $c$  in the grid such that the entry in  $c$  is both strictly greater than the average of  $c$ 's column and strictly less than the average of  $c$ 's row.

*Proposed by Holden Mui*

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