
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 $\overrightarrow{a}=(\sin 2x,\cos 2x)$ and $\overrightarrow{b}=\left( \dfrac{\sqrt{3}}{2}, -\dfrac{1}{2} \right)$, let $f(x)= \overrightarrow{a} \cdot \overrightarrow{b}$.
$(1)$ Find the period of the function $f(x)$;
$(2)$ In $\triangle ABC$, if $f(A)= \dfrac{1}{2}$, $AB=2\sqrt{3}$, $BC=2$, find the area $S$ of $\triangle ABC$.

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