
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 $f(x)=2\sin x\cos x-2\sqrt{3}\cos^2 x+\sqrt{3}$.
$(1)$ Find the interval of increase for $f(x)$;
$(2)$ If $x\in \left[ \frac{\pi}{3}, \frac{11}{24}\pi \right]$ and the lengths of two sides of an acute triangle $ABC$ are the maximum and minimum values of the function $f(x)$, with the radius of the circumcircle of triangle $ABC$ being $\frac{3}{4}\sqrt{2}$, find the area of $\triangle ABC$.

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