
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 $\{a\_n\}$ with partial sum $S\_n$, where $a\_1=t$ and the point $(a_{n+1}, S_n)$ lies on the line $y = \frac{1}{2}x - 1$ for all positive integers $n$.
(1) For what value of the real number $t$ is the sequence $\{a\_n\}$ a geometric sequence? Find the general term formula for the sequence.
(2) Let $f(x) = [x]$ (where $[x]$ denotes the greatest integer not exceeding $x$). Under the conclusion of (1), let $b_n = f(\log_3{a_n}) + 1$ and $c_n = a_n + \frac{1}{b_n b_{n+2}}$. Find the partial sum $T_n$ of the sequence $\{c_n\}$.

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