
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]


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Given an arithmetic sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. The common difference $d \neq 0$, and it is known that $S\_3 + S\_5 = 50$, while $a\_1$, $a\_4$, and $a\_13$ form a geometric sequence.

(I) Find the general term formula for the sequence $\{a\_n\}$;
(II) If sequence $\{b\_n\}$ satisfies $\frac{b\_1}{3} + \frac{b\_2}{3^2} + \dots + \frac{b\_n}{3^n} = a\_n - 1 (n \in \mathbb{N}^*)$, find the sum of the first $n$ terms for the sequence $\{nb\_n\}$, denoted as $T\_n$.

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