
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
Peter wrote  $100$  distinct integers on a board. Basil needs to fill the cells of a table  $100\times{100}$  with integers so that the sum in each rectangle  $1\times{3}$  (either vertical, or horizontal) is equal to one of the numbers written on the board. Find the greatest  $n$  such that, regardless of numbers written by Peter, Basil can fill the table so that it would contain each of numbers  $(1,2,...,n)$  at least once (and possibly some other integers).

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