
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 that $f(x)=\frac{\sqrt{3}}{2}\sin 2x+\cos^{2}x-\frac{1}{2}$, where $x\in R$.

(1) Find the smallest positive period of the function $f(x)$ and its maximum value in the interval $[0,\frac{\pi }{2}]$.

(2) If $f(x_{0})=\frac{1}{3}\cdot x_{0}\in [\frac{\pi }{6},\frac{5\pi }{12}]$, find the value of $\sin 2x_{0}$.

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