
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 a common ratio not equal to 1, the product of its first 5 terms equals 243, and $2a_3$ is the arithmetic mean of $3a_2$ and $a_4$.
(1) Find the general term formula $a_n$ for the sequence $\{a_n\}$;
(2) If the sequence $\{b_n\}$ satisfies $b_n = b_{n-1} \cdot \log_{3}a_{n+2}$ (for $n \geq 2$ and $n \in \mathbb{N}^*$), and $b_1 = 1$, find the sum $S_n$ of the first $n$ terms of the sequence $\left\{ \frac {(n-1)!}{b_{n+1}} \right\}$.

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