
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 geometric sequence $\{a_n\}$ with the first term $a_1 = \frac{1}{3}$ and a common ratio $q$ satisfying $q > 0$ and $q \neq 1$, it is also known that $a_1$, $5a_3$, and $9a_5$ form an arithmetic sequence;
(1) Find the general formula for the sequence $\{a_n\}$;
(2) Let $b_n = \log_{3} \frac{1}{a_n}$, and define $T_n = \frac{1}{b_1b_2} + \frac{1}{b_2b_3} + \frac{1}{b_3b_4} + \dots + \frac{1}{b_nb_{n+1}}$. Is there a largest integer $m$ such that for any $n \in \mathbb{N^*}$, it holds that $T_n > \frac{m}{16}$? If such an $m$ exists, find it; if it does not exist, explain why.

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