
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 $k$ is a positive number and the function $f(x)$ is an odd function over the set of real numbers $\mathbf{R}$, if the solution set of the inequality $f(x) > kx$ is $\{x \mid x < -4 \text{ or } 0 < x < 4\}$, determine the solution set of the inequality $f(|x|) \leq k|x|$.

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