
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 $f(x)=2\sqrt{3}\sin x \cos x + 2\cos^{2}x - 1$ where $x \in \mathbb{R}$.  
$(1)$ Find the maximum and minimum values of the function $f(x)$ on the interval $[0, \frac{\pi}{2}]$.  
$(2)$ If $f(x_{0}) = \frac{6}{5}$ and $x_{0} \in [\frac{\pi}{4}, \frac{\pi}{2}]$, find the value of $\cos 2x_{0}$.

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