
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 angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. It is known that $a\left(\sqrt{3}\tan B - 1\right) = \frac{b\cos A}{\cos B} + \frac{c\cos A}{\cos C}$.  
(1) Determine the size of angle $C$;  
(2) If the perimeter of the triangle is 20 and the area is $10\sqrt{3}$, with the condition $a > b$, find the lengths of the sides of the triangle.

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