
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 sequence of positive terms $\{a_{n}\}$ with $a_{1}=3$, $a_{3}=243$, and $\lg a_{n+1}+\lg a_{n-1}=2\lg a_{n}$ for $n\geqslant 2$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ Let $b_{n}=\log _{3}a_{n}$. Find the sum $S_{n}$ of the first $n$ terms of the sequence $\{a_{n}\cdot b_{n}\}$.

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