
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 that $k > 0$ is a real number, the function $f(x) = \log_a(x - 1 + k) - \log_a(1 - x), (a > 0, a \neq 1)$.

(1) Find the domain and monotonicity of $f(x)$;

(2) Determine if there exists a real number $k$ such that $f(x)$ is an odd function. If so, find $k$; if not, explain why;

(3) Solve the inequality $f(x) > 0$ with respect to $x$.

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