
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 $S\_n = a\_n + n^2 - 1 (n \in \mathbb{N}^*)$.
(I) Find the general term formula of the sequence $\{a\_n\}$;
(II) Define $x=[x] + \langle x \rangle$, where $[x]$ is the integer part of the real number $x$, and $\langle x \rangle$ is the fractional part of $x$, with $0 \leqslant \langle x \rangle < 1$. Let $c\_n = \left\langle \frac{a\_n a_{n+1}}{S\_n} \right\rangle$, 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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