
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
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $2\cos ^{2} \frac{A}{2}+\left(\cos B- \sqrt{3}\sin B \right)\cos C=1$.

$(1)$ Find the value of angle $C$;

$(2)$ If $c=2$ and the area of $\triangle ABC$ is $\sqrt{3}$, find $a$ and $b$.

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