
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 functions $f(x) = \log_a(1+x)$ and $g(x) = \log_a(1-x)$ where $(a>0 \text{ and } a \neq 1)$, let $h(x) = f(x) - g(x)$.

(1) Find the domain of function $h(x)$, determine the parity of $h(x)$, and explain the reason;  
(2) If $f(3) = 2$, find the set of $x$ for which $h(x) < 0$ holds;  
(3) If $x \in \left[0, \frac{1}{2}\right]$ and the range of function $h(x)$ is $[0, 1]$, find the range of the real number $a$.

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