
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 the sequence $\{a\_n\}$ satisfies $\frac{1}{a_{n+1}} - \frac{1}{a_n} = 2$, and $a_1 = \frac{1}{2}, n \in \mathbb{N}_+$.
(I) Let the sequence $\{b_n\}$'s first $n$ terms sum be $S_n$. If the sequence $\{b_n\}$ satisfies $b_n = \begin{cases} \frac{1}{\sqrt{n-1} + \sqrt{n+1}} & (n = 2k - 1) \\ a_{\frac{n}{2}} a_{\frac{n}{2} + 1} & (n = 2k) \end{cases} (k \in \mathbb{N}_+)$, find $S_{64}$;
(II) Let $T_n = \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + \ldots + \frac{1}{a_n}$, does there exist a constant $c$ such that $\{\frac{T_n}{n + c}\}$ is an arithmetic sequence? Please explain the reason.

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