
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 $\{a_n\}$ be a positive geometric sequence with $a_1=2$, and $\frac{1}{2}a_3$ is the arithmetic mean of $3a_1$ and $2a_2$.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) If the terms of the sequence $\{b_n\}$ are positive, and $b_n$ is the geometric mean of $\frac{n}{a_n}$ and $\frac{n}{a_{n+2}}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$. If for any $n\in\mathbb{N}^*$, $T_n > \log_m 2$ holds, find the range of the real number $m$.

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