
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 $f\left( x \right)=\sin\left( 2x-\frac{\pi }{6} \right)+2\cos^2x-2$.

(1) Find the interval(s) where $f\left( x \right)$ is increasing;

(2) In triangle $ABC$, let the sides opposite to angles $A$, $B$, $C$ be $a$, $b$, $c$ respectively. Given $c=\sqrt{3}$, $2f\left( C \right)=-1$, and $2\sin A=\sin B$, find the area of triangle $ABC$.

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