
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 functions $f(x)=\log_a(x+1)$ and $g(x)=\log_a(x-1)$ where $a>0$ and $a \neq 1$,
(1) Determine the parity of the function $f(x) + g(x)$ and explain the reasoning;
(2) Find the set of values for $x$ that satisfies $f(x) - g(2x) > 0$.

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