
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 $f(x)=2\sin (\frac{\pi }{2}+x)\sin (\frac{\pi }{3}+x)$, $x∈R$.

(I) Find the smallest positive period of the function $f(x)$, and write out the interval where $f(x)$ is monotonically increasing;

(II) Given ${{x}_{0}}∈ (\frac{\pi }{6},\frac{\pi }{2})$, and $f({{x}_{0}})=\frac{4}{5}+\frac{\sqrt{3}}{2}$, find the value of $f({{x}_{0}}+\frac{\pi }{6})$.

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