
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 vectors $\overrightarrow{m}=(\sin ^{2}x+ \frac{1+\cos 2x}{2},\sin x)$, $\overrightarrow{n}=( \frac{1}{2}\cos 2x- \frac{ \sqrt{3}}{2}\sin 2x,2\sin x)$, define the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$, $x\in \mathbb{R}$.

(I) Find the smallest positive period of the function $f(x)$;

(II) If $x\in [0, \frac{\pi}{2}]$, find the range of the function $f(x)$.

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