
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
In a geometric sequence $\{a_{n}\}$, $a_{1}=3$, and $3a_{3}$ is the arithmetic mean of $a_{4}$ and $a_{5}$.  
$(1)$ Find the general formula for $\{a_{n}\}$.  
$(2)$ If $a_{n} \gt 0$, ${b}_{n}=\frac{n{a}_{n}}{3}$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$ denoted by $S_{n}$.

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