
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 the function $f(x)=2\sin x\sin \left(x+\dfrac{\pi }{6}\right)$.

$(1)$ When $x\in \left[0,\dfrac{\pi }{2}\right]$, find the range of the function $f(x)$.

$(2)$ In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $f(A)=\sqrt{3}$ and $A$ is an acute angle, and the area $S=2\sqrt{3}$, find the minimum value of the side length $a$.

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