
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 \( f(x) = \log_a \frac{x+b}{x-b} \) where \(a > 1\) and \(b > 0\),

(1) Find the domain of \( f(x) \).

(2) Determine the parity (even or odd nature) of \( f(x) \).

(3) Analyze the monotonicity of \( f(x) \).

(4) Find the inverse function of \( f(x) \).

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