
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 that the function $f(x)$ is an even function defined on $(-\infty, 0) \cup (0, +\infty)$, and when $x > 0$, the graph of $f(x) = \log_{a}x$ passes through the point $(5, 2)$.
$(1)$ Find the value of $a$.
$(2)$ Find the analytical expression of $f(x)$.
$(3)$ Find the solution set of the inequality $f(x) > 4$.

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