
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 positive integer. For a positive integer  $m$ , we partition the set  $\{1, 2, 3,...,m\}$  into  $n$  subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of  $n$ , find the largest positive integer  $m$  for which such a partition exists.

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