
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 odd function $f(x)$ is defined on the domain $(-\infty, 0) \cup (0, +\infty)$, and $f'(x)$ is its derivative function, which satisfies the following conditions: $(1)$ When $x > 0$, $f'(x) < \frac{3f(x)}{x}$; $(2)$ $f(1) = \frac{1}{2}$; $(3)$ $f(2x) = 2f(x)$. Determine the solution set of the inequality $\frac{f(x)}{4x} < 2x^2$.

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