
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  $ P_n\equal{}\sqrt[n]{\frac{(3n)!}{(2n)!}}\ (n\equal{}1,\ 2,\ \cdots)$ .
Find the following limits.

(1)  $ \lim_{n\to\infty} \frac{P_n}{n}$ .

(2)  $ \lim_{n\to\infty} \left(\frac{n\plus{}2}{n}\right)^{P_n}$ .

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