
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 $f(x) = 6\cos^2x - \sqrt{3} \sin 2x$,
(1) Find the maximum value of $f(x)$ and its minimum positive period;
(2) If acute angle $\alpha$ satisfies $f(\alpha) = 3 - 2\sqrt{3}$, find the value of $\tan\left(\frac{4}{5}\alpha\right)$.

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