
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 sum of the first $n$ terms denoted as $S\_n$. It is known that $S\_n=n(n+1)$, where $n \in \mathbb{N}^+$.

(1) Find the general term formula of the sequence $\{a\_n\}$.
(2) If a sequence $\{b\_n\}$ satisfies $a\_n = \frac{b\_1}{3+1} + \frac{b\_2}{3^2+1} + \frac{b\_3}{3^3+1} + \dots + \frac{b\_n}{3^n+1}$, find the general term formula of the sequence $\{b\_n\}$.
(3) Define a sequence $\{c\_n\} = \frac{a\_n b\_n}{4}$, where $n \in \mathbb{N}^+$. Find the sum of the first $n$ terms of the sequence $\{c\_n\}$, denoted as $T\_n$.

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