
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 sequence of positive numbers with the sum of its first $n$ terms denoted as $S\_n$. For all $n \in \mathbb{N}^+$, it is given that $8S\_n = (a\_n + 2)^2$.

(1) Write out the first three terms of the sequence $\{a\_n\}$;

(2) Find the general term formula for $\{a\_n\}$ (provide a proof process);

(3) Let $b\_n = \frac{4}{a\_n \cdot a\_{n+1}}$, and $T\_n$ be the sum of the first $n$ terms of the sequence $\{b\_n\}$. Find the smallest positive integer $m$ such that $T\_n < \frac{m}{20}$ holds for all $n \in \mathbb{N}^+$.

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