
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
Let $f(x) = 2a^{x} - 2a^{-x}$ where $a > 0$ and $a \neq 1$. <br/> $(1)$ Discuss the monotonicity of the function $f(x)$; <br/> $(2)$ If $f(1) = 3$, and $g(x) = a^{2x} + a^{-2x} - 2f(x)$, $x \in [0,3]$, find the minimum value of $g(x)$.

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