
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 an arithmetic sequence $\{a_n\}$ with first term $a_1 = \frac{1}{3}$ and common ratio $q$ such that $q > 0$ and $q \neq 1$. It is also known that $(a_1, 5a_3, 9a_5)$ forms an arithmetic sequence.

(1) Find the general term of the sequence $\{a_n\}$.

(2) Let $b_n = \log_3 \frac{1}{a_n}$. Define $T_n = \frac{1}{b_1b_2} + \frac{1}{b_2b_3} + \frac{1}{b_3b_4} + \cdots + \frac{1}{b_n b_{n+1}}$. Determine if there exists a maximum integer $m$ such that $T_n > \frac{m}{16}$ holds for any $n \in \mathbb{N}^*$. If it exists, find $m$; otherwise, explain the reason.

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