
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 that $a$, $b$, $c$ are the opposite sides of angles $A$, $B$, $C$ in triangle $\triangle ABC$, and $2b=c+2a\cos C$.  
$(1)$ Find the value of angle $A$.  
$(2)$ If $\cos B=\frac{\sqrt{3}}{3}$, find the value of $\sin \left(2B-A\right)$.  
$(3)$ If the area of $\triangle ABC$ is $\frac{10\sqrt{3}}{3}$ and $a=3$, find the perimeter of $\triangle ABC$.

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