
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 an exponential sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, with $a_1=1$, and $S_1$, $2S_2$, $3S_3$ forming an arithmetic sequence.

(1) Find the general formula for the sequence $\{a_n\}$;
(2) Define $\frac{1}{b_n} = \log_3{a_{n+1}} \cdot \log_3{a_{n+2}}$. Find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.

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