
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 $\overrightarrow{m}=(2\cos x,y-2\sqrt{3}\sin x\cos x)$ and $\overrightarrow{n}=(1,\cos x)$, with $\overrightarrow{m}$ parallel to $\overrightarrow{n}$,
(I) Express $y$ as a function of $x$, $f(x)$, and find the interval where $f(x)$ is strictly increasing;
(II) Let $a$, $b$, and $c$ represent the lengths of sides $A$, $B$, and $C$ of $\triangle ABC$, respectively. If $f\left(\frac{c}{2}\right)=3$, $c=2\sqrt{6}$, and $a+b=6$, find the area of $\triangle ABC$.

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