
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 $f(x)=2\sin \omega x\cos \omega x-2\sqrt{3}\cos^{2}\omega x+\sqrt{3}$ ($\omega > 0$), and the distance between two adjacent axes of symmetry of the graph of $y=f(x)$ is $\frac{\pi}{2}$.

(Ⅰ) Find the interval of monotonic increase for the function $f(x)$;

(Ⅱ) Given that in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, angle $C$ is acute, and $f(C)=\sqrt{3}$, $c=3\sqrt{2}$, $\sin B=2\sin A$, find the area of $\triangle ABC$.

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