
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 $f(x)=\log_{a} \frac{1-mx}{x-1}$ ($a > 0$ and $a \neq 1$) is an odd function.  
$(1)$ Find the value of the real number $m$;  
$(2)$ Determine the monotonicity of the function $f(x)$ on the interval $(1,+\infty)$ and explain the reason;  
$(3)$ When $x \in (n, a-2)$, the range of the function $f(x)$ is $(1, +\infty)$. Find the values of the real numbers $n$ and $a$.

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