
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 sequence of positive terms $\{a\_n\}$ with the sum of the first $n$ terms denoted as $S\_n$. The first term is $a\_1=1$, and the point $P(a\_n, a_{n+1}^2)$ lies on the curve $y=x^2+4x+4$.
(1) Find $a\_n$ and $S\_n$;
(2) If sequence $\{b\_n\}$ satisfies $b\_1=17$ and $b_{n+1}-b\_n=2n$, find the value of $n$ that minimizes $\frac{b\_n}{\sqrt{S\_n}}$.

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