
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 $f(x)$ satisfies the condition $f(ax-1) = \lg \frac{x+2}{x-3}$ where $a \neq 0$,
(1) Find the expression for $f(x)$;
(2) Find the domain of $f(x)$;
(3) Does there exist a real number $a$ such that $f(x)$ is an odd or even function? If it exists, find the value of $a$; otherwise, explain why not.

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