
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
Consider a chessboard that is infinite in all directions. Alex the T-rex wishes to place a positive integer in each square in such a way that:


- No two numbers are equal.
- If a number  $m$  is placed on square  $C$ , then at least  $k$  of the squares orthogonally adjacent to  $C$  have a multiple of  $m$  written on them.

What is the greatest value of  $k$  for which this is possible?

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