
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, and it satisfies $\sqrt{3}a\cos C - c\sin A = 0$.
$(1)$ Find the size of angle $C$.
$(2)$ Given $b=6$ and the area of $\triangle ABC$ is $6\sqrt{3}$, find the value of side length $c$.

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