
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 a sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$, and the point $P\_n(n, S\_n)$ ($n \in \mathbb{N}^*$) lies on the curve $f(x) = \frac{3}{2}x^2 - \frac{1}{2}x$. Sequence $\{b\_n\}$ is a geometric sequence that satisfies $b\_1 = a\_1 + 1$ and $b\_2 = a\_3 - 1$.

(I) Find the general term formulas for sequences $\{a\_n\}$ and $\{b\_n\}$.
(II) Let $c\_n = (-1)^n a\_n + b\_n$, find the sum of the first $n$ terms, $T\_n$, for sequence $\{c\_n\}$.

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