
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 $I_{n}=\{1,2,3,\ldots,n\}$, and $P_{n}=\left\{ \frac{m}{\sqrt{k}} \mid m\in I_{n}, k\in I_{n}\right\}$.  
(1) Find the number of elements in the set $P_{7}$.  
(2) If the subset $A$ of $P_{n}$ has the property that the sum of any two of its elements is not the square of an integer, then $A$ is called a "sparse set". Find the maximum value of $n$ such that $P_{n}$ can be partitioned into the union of two disjoint sparse sets.

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