
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 $\overrightarrow{m}=(2\cos x,y-2 \sqrt{3}\sin x\cos x)$ and $\overrightarrow{n} =(1,\cos x)$, and $\overrightarrow{m} \parallel \overrightarrow{n}$,

(Ⅰ) Try to express $y$ as a function $f(x)$ of $x$, and find the interval of monotonic increase for $f(x)$;  
(Ⅱ) Given that $a$, $b$, $c$ are the lengths of the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$ respectively, if $f\left( \frac{c}{2} \right)=3$ and $c=2 \sqrt{6}$, $a+b=6$, find the area of $\triangle ABC$.

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