
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 $\{a_n\}$ is a geometric sequence with the first term being $1$ and is monotonically increasing, and it satisfies that $a_3$, $\frac{5}{3}a_4$, $a_5$ form an arithmetic sequence.  
(1) Find the general formula for $\{a_n\}$.  
(2) If $b_n=\log_{3}(a_n\cdot a_{n+1})$ $(n\in\mathbb{N}^*)$, find the sum of the first $n$ terms $S_n$ of the sequence $\{a_n\cdot b_n\}$.

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