
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 $f(x)=2\sqrt{3}\sin^2x+2\sin x\cos x-\sqrt{3}$, where $x\in \left[\frac{\pi}{3},\frac{11\pi}{24}\right]$.

$(1)$ Find the range of the function $f(x)$;

$(2)$ Given that the lengths of two sides of an acute triangle $\Delta ABC$ are the maximum and minimum values of the function $f(x)$, and the radius of the circumscribed circle of $\Delta ABC$ is $\frac{3\sqrt{2}}{4}$, find the area of $\Delta ABC$.

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