
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 all terms in the sequence $\{a_n\}$ are positive numbers, and the sum of the first $n$ terms, denoted as $S_n$, satisfies the equation $2S_n = (a_n + 3)(a_n - 2)$ for all $n \in \mathbb{N}^*$.

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

(2) Find the sum of the first $n$ terms, denoted as $T_n$, for the sequence $\{\frac{1}{a_{2n-1} \cdot a_{2n+1}}\}$.

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