
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 that the sum of the first $n$ terms of the sequence ${a\_n}$ is $S\_n$, and $S\_n = n(n+1) (n \in \mathbb{N}^*)$.
(I) Find the general term formula of the sequence ${a\_n}$;
(II) If the 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 ${b\_n}$;
(III) Let $c\_n = \frac{a\_n b\_n}{4} (n \in \mathbb{N}^*)$, 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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