
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 a function defined on $(0, \frac{\pi}{2})$, let $f'(x)$ be its derivative such that $\frac{f(x)}{\sin x} < \frac{f'(x)}{\cos x}$ always holds. Derive the inequality that compares the function values at $f(\frac{\pi}{2})$ and $f(\frac{\pi}{6})$.

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