
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 $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, and $2S_n=3a_n-2$ ($n\in\mathbb{N}^*$).  
$(1)$ Find $a_n$ and $S_n$;  
$(2)$ If $b_n=\log_{3}(S_n+1)$, find the sum of the first $n$ terms of the sequence $\{b_{2n}\}$, denoted as $T_n$.

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