
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 $S\_n = 3^n$, and the sequence $\{b\_n\}$ satisfies $b\_1 = -1$, $b_{n+1} = b\_n + (2n - 1)$ ($n \in \mathbb{N}^*\$).
(I) Find the general term formula $a\_n$ for the sequence $\{a\_n\}$;
(II) Find the general term formula $b\_n$ for the sequence $\{b\_n\}$;
(III) If $c\_n = \frac{a\_n \cdot b\_n}{n}$, find the sum of the first $n$ terms $T\_n$ for the sequence $\{c\_n\}$.

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