
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 $\overrightarrow{a}=(2\cos\omega x,-2)$ and $\overrightarrow{b}=(\sqrt{3}\sin\omega x+\cos\omega x,1)$, where $\omega\ \ \gt 0$, and the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}+1$. The distance between two adjacent symmetric centers of the graph of $f(x)$ is $\frac{\pi}{2}$.  
$(1)$ Find $\omega$;  
$(2)$ Given $a$, $b$, $c$ are the opposite sides of the three internal angles $A$, $B$, $C$ of scalene triangle $\triangle ABC$, and $f(A)=f(B)=\sqrt{3}$, $a=\sqrt{2}$, find the area of $\triangle ABC$.

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