
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
Let $f(x)$ be an odd function defined on $(-\pi, 0) \cup (0, \pi)$ with its derivative denoted as $f'(x)$. It is given that $f\left( \frac{\pi}{2} \right) = 0$. If for $x \in (0, \pi)$, the inequality $f'(x)\sin x - f(x)\cos x < 0$ holds, determine the solution set for the inequality $f(x) < 2f\left( \frac{\pi}{6} \right)\sin x$.

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