
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 that in the positive geometric sequence $\{a\_n\}$, $a\_1=3$, and $\frac{1}{2}a\_3$ is the arithmetic mean of $9a\_1$ and $8a\_2$.
(1) Find the general term formula of the sequence $\{a\_n\}$;
(2) Let $b\_n=\frac{1}{\log\_2 a\_n \cdot \log\_3 a\_{n+1}}$, find the sum of the first $n$ terms of the sequence $\{b\_n\}$, denoted as $T\_n$; and if $T\_n > \log\_m 2$ holds for any $n \in \mathbb{N}^*$, find the range of the real number $m$.

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