
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 $f\left(x\right)=\sin x+\sqrt{3}\cos x$.  
$(1)$ Find the interval where the function $y=f\left(x\right)$ is monotonically increasing.  
$(2)$ Let $\alpha$ and $\beta$ ($\alpha \gt \beta$) be the two solutions of the equation $f\left(x\right)=\frac{1}{2}$ in the interval $[-\frac{π}{3},\frac{{5π}}{3})$. Find the value of $\cos \left(\alpha -\beta \right)$.  
$(3)$ In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $c=\sqrt{3}$, $f\left(C\right)=0$, and $\sin A+\sin B=2\sqrt{10}\sin A\sin B$, find the area of triangle $ABC$.

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