
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 $f\left(x\right)=2 \sqrt{3}\sin x\cos x+2{\cos }^{2}x-1\left(x∈R\right)$

(I) Find the smallest positive period of the function $f(x)$ and its maximum and minimum values in the interval $\left[0, \frac{π}{2}\right]$;

(II) If $f\left({x}_{0}\right)= \frac{6}{5},{x}_{0}∈\left[ \frac{π}{4}, \frac{π}{2}\right]$, find the value of $\cos 2{x}_{0}$.

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