
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 $f(x)=\cos\left(2x-\frac{\pi}{6}\right)-\sqrt{3}\cos(2x)-\frac{1}{2}$.  
$(1)$ Find the smallest positive period of $f(x)$ and the center of symmetry of its graph;  
$(2)$ If $x_0\in\left[\frac{5\pi}{12},\frac{2\pi}{3}\right]$ and $f(x_0)=\frac{\sqrt{3}}{3}-\frac{1}{2}$, find the value of $\cos(2x_0)$.

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