
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 function \( f(x)=\left(x^{2}-4\right)^{-\frac{1}{2}}(x>2), f^{-1}(x) \) is the inverse function of \( f(x) \).

(1) If the sequence \(\{a_{n}\}\) is defined by \(a_{1}=1\) and \(\frac{1}{a_{n+1}}=f^{-1}\left(a_{n}\right) \left(n \in \mathbf{N}_{+}\right)\), find \(a_{n}\).

(2) Let \(S_{n}=a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}\) and \(b_{n}=S_{2n+1}-S_{n}\). Does there exist a maximum positive integer \(p\) such that \(b_{n}<\frac{1}{p}\) holds for all \(n \in \mathbf{N}_{+}\)? Provide an explanation or proof.

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