
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
Let $\{a_n\}$ be a positive arithmetic sequence with the sum of the first $n$ terms denoted as $S_n$, and it satisfies $a_2+a_5=7$ and $a_3 a_4=12$.
$(1)$ Find the general formula for the $n$-th term of $\{a_n\}$.
$(2)$ If $b_n = \frac{a_{n+1}}{a_{n+2}} + \frac{a_{n+2}}{a_{n+1}} - 2$, calculate the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.

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