
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 the sequence $\{a_n\}$ with its first $n$ terms sum $S_n$, and $a_1=1$, $S_n=2a_{n+1}$.
1. Find the general term formula for the sequence $\{a_n\}$;
2. When $b_{n}=\log_{\frac{3}{2}}(3a_{n+1})$, find the sum of the first $n$ terms $T_n$ for the sequence $\{\frac{1}{b_{n}b_{n+1}}\}$.

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