
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]


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Given a function $f(x) = a^x + \frac{1}{a^x}$ defined on $\mathbb{R}$, where $a > 1$:
(1) Determine the parity and monotonicity of $f(x)$, and provide the reasoning;
(2) Solve the inequality: $f(x - 1) > f(2x + 1)$.

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