
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
Integers are written in the cells of a table  $2010 \times 2010$ . Adding  $1$  to all the numbers in a row or in a column is called a *move*. We say that a table is *equilibrium* if one can obtain after finitely many moves a table in which all the numbers are equal.
[list=a]
[*]Find the largest positive integer  $n$ , for which there exists an *equilibrium* table containing the numbers  $2^0, 2^1, \ldots , 2^n$ .
[*] For this  $n$ , find the maximal number that may be contained in such a table.
[/list]

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