
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
Consider a sequence $\{a\_n\}$ with its first $n$ terms sum denoted as $S\_n$, satisfying $a\_1=1$, $2S\_n=a_{n+1}-1$.

(1) Find the general formula for the $n$-th term of the sequence.

(2) Let $b\_n=\log\_3 a_{n+1}$. Denote the first $n$ terms sum of sequence $\{b\_n\}$ as $T\_n$. Find the sum of the first $n$ terms of the sequence $\left\{\frac{1}{T\_n} + 4a\_n\right\}$.

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