
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 vectors $\overrightarrow{a}=(\sin \omega x+\cos \omega x,\sqrt{3}\cos \omega x)$, $\overrightarrow{b}=(\cos \omega x-\sin \omega x,2\sin \omega x)(\omega > 0)$, if the distance between the adjacent symmetrical axes of the function $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$ is equal to $\frac{\pi }{2}$.

(I) Find the value of $\omega$;
(II) In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $f(C)=1$, $c=2$, and $\sin C+\sin (B-A)=3\sin 2A$, find the area of $\triangle ABC$.

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