
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 $\overrightarrow{a}=(\cos x,-\frac{1}{2})$, $\overrightarrow{b}=(\sqrt{3}\sin x,\cos 2x)$, where $x \in \mathbb{R}$, let the function $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$

$(1)$ Find the smallest positive period of $f(x)$;

$(2)$ Find the maximum and minimum values of $f(x)$ in the interval $\left[ 0,\frac{\pi }{2} \right]$.

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