
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 is given that $(2a-b)\cdot\cos C=c\cdot\cos B$.  
$(1)$ Find the magnitude of angle $C$;  
$(2)$ If $c=2$ and the area of $\triangle ABC$ is $\sqrt{3}$, find the perimeter of this triangle.

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