
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 $f(x)=\log \frac{1-ax}{\frac{1}{2}x-1}$, where $a$ is a constant, and the graph of $f(x)$ is symmetric about the origin.  
$(1)$ Find the value of $a$;  
$(2)$ When $x \in (1, +\infty)$, $f(x)+\log \frac{1}{2} (x-1) < m$ always holds, find the range of the real number $m$;  
$(3)$ If the equation $f(x)=\log \frac{1}{2} (x+k)$ has a solution in the interval $[2,3]$, find the range of $k$.

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