
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 the function $f(x) = 2\sin x\cos x + 2\sqrt{3}\cos^2x - \sqrt{3}$.
 (1) Find the smallest positive period of the function $f(x)$;
(2) If $f(x_0 - \frac{\pi}{12}) = \frac{6}{5}$, where $x_0 \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right]$, find the value of $\cos 2x_0$.

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