
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
On a square table of  $2011$  by  $2011$  cells we place a finite number of napkins that each cover a square of  $52$  by  $52$  cells. In each cell we write the number of napkins covering it, and we record the maximal number  $k$  of cells that all contain the same nonzero number. Considering all possible napkin configurations, what is the largest value of  $k$ ?

*Proposed by Ilya Bogdanov and Rustem Zhenodarov, Russia*

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