
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
A sequence $\{a\_n\}$ has positive terms, and its partial sum $S\_n$ satisfies the equation $2S\_n^2 - (3n^2 + 3n - 2)S\_n - 3(n^2 + n) = 0$, where $n \in \mathbb{N}^*$.

(I) Find the value of $a\_1$.
(II) Find the general term formula for the sequence $\{a\_n\}$.
(III) Let $b\_n = \frac{a\_n}{3^{n+1}}$, find the partial sum $T\_n$ for the sequence $\{b\_n\}$.

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