
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 a sequence $\{a_n\}$ with its first $n$ terms sum $S_n$, it is known that $a_1 = 1$ and $S_{n+1} - 3S_n = 1$ for all positive integers $n$.
(1) Find the general formula for the $n$-th term of the sequence $\{a_n\}$.
(2) If a new sequence $\{b_n\}$ satisfies $b_n = \log_3 a_{n+1}$ and its first $n$ terms sum is $T_n$, find the value of $\frac{1}{T_1} + \frac{1}{T_2} + \dots + \frac{1}{T_{100}}$.

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