
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 an increasing arithmetic sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. The first term $a\_1 = 1$, and the sequence $a\_2 + 1$, $a\_4 + 1$, $S\_4$ forms a geometric progression.

(I) Find the general term formula for the sequence $\{a\_n\}$.
(II) Let $b\_n = \frac{a\_n}{a_{n+1}} + \frac{a_{n+1}}{a\_n} - 2$. Find 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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