
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 $O$ is the coordinate origin, $\overrightarrow{OA}=({2\cos x,\sqrt{3}})$, $\overrightarrow{OB}=({\sin x+\sqrt{3}\cos x,-1})$, if $f(x)=\overrightarrow{OA} \cdot \overrightarrow{OB}+2$.
$(1)$ Find the smallest positive period of the function $f(x)$;
$(2)$ When $x \in (0,\frac{\pi}{2})$, find the range of the function $f(x)$.

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