
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
Let $a, b \in \mathbb{R}$, and $a \neq 1$. Suppose the odd function $f(x) = \lg \frac{1+ax}{1+x}$ is defined on the interval $(-b, b)$.  
$(1)$ Find the value of $a$;  
$(2)$ Determine the range of values for $b$;  
$(3)$ Solve the inequality $f(x) > 0$.

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