
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 triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. Let $\overrightarrow{m}=\left(b+c,\sin A\right)$ and $\overrightarrow{n}=\left(a+b,\sin C-\sin B\right)$ be vectors, and $\overrightarrow{m}$ is parallel to $\overrightarrow{n}$.  
$(1)$ Find angle $C$.  
$(2)$ If $b=4$ and the area of $\triangle ABC$ is $4\sqrt{3}$, find the perimeter of $\triangle ABC$.

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