
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 f(x) is an odd function defined on (-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}), its derivative is f'(x), and when x \in (0, \frac{\pi}{2}), f'(x) \tan x - \frac{f(x)}{\cos^2 x} > 0. Also, f(\frac{\pi}{4}) = 0. Find the range of x that satisfies the inequality f(x) < \sqrt{3} f(\frac{\pi}{6}) \tan x.

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