
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
The differentiable function \( f:\left(0, \frac{\pi}{2}\right) \rightarrow \mathbf{R} \) satisfies \( f'(x) \tan x \geqslant (2 \sin x - 1) f(x) \). Given that \( f\left(\frac{\pi}{6}\right) = 1 \), find the minimum value of \( f\left(\frac{\pi}{3}\right) \).

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