
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
In the arithmetic sequence $\{a\_n\}$, $a\_1 + a\_3 = 10$, and the common difference $d = 3$. Let $b\_n = \frac{1}{a\_n a\_{n+1}}$, and the sum of the first $n$ terms of the sequence $\{b\_n\}$ is denoted as $T\_n$.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. Find the sum of the first $n$ terms, $T\_n$, for the sequence $\{b\_n\}$.
3. Determine if there exist positive integers $m$ and $n$ ($1 < m < n$) such that $T\_1$, $T\_m$, and $T\_n$ form a geometric sequence. If so, find all possible values of $m$ and $n$; if not, explain the reason.

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