
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
For a positive integer \( n \), let \( \mathrm{I}_{\mathrm{n}}=\{1,2, \ldots, \mathrm{n}\} \) and \( \mathrm{P}_{\mathrm{n}}=\left\{\left.\frac{\mathrm{m}}{\sqrt{k}} \right| \mathrm{m} \in \mathrm{I}_{\mathrm{n}}, \mathrm{k} \in \mathrm{I}_{\mathrm{n}}\right\} \). A subset \( \mathrm{A} \) of \( \mathrm{P}_{\mathrm{n}} \) is called a "sparse set" if the sum of any two elements in \( \mathrm{A} \) is not a perfect square. Determine the maximum value of \( n \) such that \( \mathrm{P}_{\mathrm{n}} \) can be partitioned into the union of two disjoint sparse sets.

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