
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 $\triangle ABC$ be a triangle with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. Given that $c\sin \left(A-B\right)=b\sin \left(C-A\right)$.<br/>$(1)$ If $a^{2}=bc$, find the value of angle $A$;<br/>$(2)$ If $a=2$ and $\cos A=\frac{4}{5}$, find the perimeter of $\triangle ABC$.

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