
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]


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Given a geometric sequence $\{a_n\}$ with a common ratio not equal to $1$, the product of the first $5$ terms is $243$, and $2a_3$ is the arithmetic mean of $3a_2$ and $a_4$.  
$(1)$ Find the general formula for the term $a_n$ of the sequence $\{a_n\}$;  
$(2)$ If the sequence $\{b_n\}$ satisfies $b_n=b_{n-1}\cdot\log_3a_{n+2}$ $(n\geqslant 2$ and $n\in\mathbb{N}^*)$ and $b_1=1$, find the sum of the first $n$ terms $S_n$ of the sequence $\left\{ \frac {(n-1)!}{b_{n+1}}\right\}$.

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