
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
The sum of the first $n$ terms of a sequence $\{a_{n}\}$ is $S_{n}$, given $a_{1}=2$, $2S_{n}=a_{n+1}-2$.  
$(1)$ Find $\{a_{n}\}$;  
$(2)$ If ${b_n}={\log_3}\frac{{a_n}}{2}$, find the sum of the first $n$ terms of $\{a_{n}\cdot b_{n}\}$, denoted as $T_{n}$.

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