
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 non-constant arithmetic sequence $\{a\_n\}$ with distinct terms, the sum of its first four terms is $16$. Additionally, $a\_1$, $a\_2$, and $a\_5$ form a geometric sequence. The sequence $\{b\_n\}$ is defined as $b\_n= \frac {1}{a\_n a_{n+1}}$.

(I) Find the general term formula $a\_n$ of the sequence $\{a\_n\}$ and the sum $T\_n$ of the first $n$ terms of the sequence $\{b\_n\}$.
(II) Determine if there exist positive integers $s$ and $t$ ($1 < s < t$) such that $T\_1$, $T\_s$, and $T\_t$ form a geometric sequence. If they exist, find the values of $s$ and $t$; if not, provide a reason.

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