
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 $ABC$, sides $a$, $b$, and $c$ are opposite to angles $A$, $B$, and $C$ respectively. Given that $2b\cos{C} = a\cos{C} + c\cos{A}$,  
(I) Find the size of angle $C$;  
(II) If $b=2$ and $c=\sqrt{7}$, find the value of $a$ and the area of triangle $ABC$.

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