
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
Let $\triangle ABC$ have sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, with $\cos C=\frac{2}{3}$ and $c=\sqrt{5}$. <br/>$(1)$ Find the circumference of the circumcircle of $\triangle ABC$; <br/>$(2)$ If $2a=3b$, find the perimeter of $\triangle ABC$.

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