
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 a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $S_n = a_n + n^2 - 1$ ($n \in \mathbb{N}^*$).

(1) Find the general formula for the sequence $\{a_n\}$.

(2) 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 $c_n = \langle \frac{a_n a_{n+1}}{S_n} \rangle$, 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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