
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 $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite angles $A$, $B$, and $C$ respectively. Given that $c\sin B + \left(a + \frac{c^2}{a} - \frac{b^2}{a}\right)\sin C = 2c\sin A$, and the radius of the circumcircle of the triangle is $\sqrt{3}$.  
$(1)$ Find the measure of angle $C$.  
$(2)$ If the area of $\triangle ABC$ is $\sqrt{3}$, find the value of $\cos 2A - 2\sin^2 B + 1$.

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