
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 $f(x) = \log_a \left(\frac{1-mx}{1+x}\right)$ ($a>0$ and $a \neq 1$, $m \neq -1$) is an odd function defined on the interval $(-1, 1)$,

(1) Find the value of $f(0)$ and the value of the real number $m$;  
(2) Determine the monotonicity of the function $f(x)$ on the interval $(-1, 1)$ and explain the reason;  
(3) If $f\left(\frac{1}{2}\right) > 0$ and $f(b-2) + f(2b-2) > 0$ hold, find the range of the real number $b$.

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