
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
Given a positive geometric sequence $\{a_n\}$, where $2a_1 + a_2 = 15$ and $a_4^2 = 9a_1a_5$,

(1) Find the general formula for the sequence $\{a_n\}$;

(2) Let $b_n = \log_3 a_1 + \log_3 a_2 + \ldots + \log_3 a_n$; the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{b_n} \right\}$ is denoted as $S_n$. Determine whether there exists a positive integer $n$ such that $S_n > \frac{39}{20}$. If it exists, find the minimum value of $n$; if not, explain why.

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