
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 $f(x) = \log_{a}(1 + x)$, $g(x) = \log_{a}(1 - x)$, where $a > 0$ and $a \neq 1$. Let $h(x) = f(x) - g(x)$.

1. Find the domain of the function $h(x)$, determine the parity (odd or even) of $h(x)$, and explain the reasons.
2. If $f(3) = 2$, find the set of $x$ that makes $h(x) < 0$.
3. If the range of function $h(x)$ is $[0, 1]$ when $x \in [0, \frac{1}{2}]$, find the value of the real number $a$.

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