
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 $f(x)=2 \sqrt{3} \sin ^{2}x+2\sin x\cos x- \sqrt{3}$, 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-angled $\triangle ABC$ are the maximum and minimum values of the function $f(x)$, and the radius of the circumscribed circle of $\triangle ABC$ is $\frac{3 \sqrt{2}}{4}$, find the area of $\triangle ABC$.

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