
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 $f(x) = \log_a(3x+1)$ and $g(x) = \log_a(1-3x)$, where $a>0$ and $a \neq 1$:
1. Find the domain of the function $F(x) = f(x) - g(x)$.
2. Determine the parity (even or odd) of $F(x) = f(x) - g(x)$ and provide the reasoning.
3. Determine for which values of $x$, $f(x) - g(x) > 0$ holds true.

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