
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 in triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, if $4\sin A - b\sin B = c\sin \left(A-B\right)$. Find the value of $a$.
$(1)$ Find the value of $a$;
$(2)$ If the area of $\triangle ABC$ is $\frac{\sqrt{3}(b^{2}+c^{2}-a^{2})}{4}$, find the maximum perimeter of $\triangle ABC$.

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