
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 that the sum of the first $n$ terms of a sequence $\{a_{n}\}$ is $S_{n}$, $a_{1}=3$, and $2S_{n}=3a_{n}-3$.  
$(1)$ Find the general formula for $\{a_{n}\}$;  
$(2)$ Let the sequence $\{b_{n}\}$ satisfy: $b_{n}=a_{n}+\log _{3}a_{n}$. If the sum of the first $n$ terms of $\{b_{n}\}$ is denoted as $T_{n}$, find $T_{n}$.

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