
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 $f(x)=\log_{a} \frac{1-mx}{x+1} (a > 0, a \neq 1, m \neq -1)$, which is defined on $(-1,1)$ as an odd function.  
$(I)$ Find the value of $f(0)$ and the value of the real number $m$;  
$(II)$ When $m=1$, determine the monotonicity of the function $f(x)$ on $(-1,1)$ and provide a proof;  
$(III)$ If $f\left( \frac{1}{2}\right) > 0$ and $f(b-2)+f(2b-2) > 0$, find the range of the real number $b$.

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