
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 triangle $ABC$ with internal angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively, satisfying $c\cos B + b\cos C = 2a\cos C$.
(1) Find the size of angle $C$;
(2) If $c=2\sqrt{3}$ and the area of $\triangle ABC=S_{\triangle ABC}=2\sqrt{3}$, find the values of $a$ and $b$.

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