
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 \( f(x)=\log_{a}\left(x+\sqrt{x^{2}-2}\right) (a > 0, a \neq 1) \) and its inverse function \( f^{-1}(x) \). Define \( g(n)=\frac{\sqrt{2}}{2} f^{-1}\left(n+\log_{a} \sqrt{2}\right) \). If \( g(n)<\frac{3^{n}+3^{-n}}{2} (n \in \mathbf{N}_{+}) \), find the range of values for \( a \).

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