
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 an acute triangle $\triangle ABC$ with internal angles $A$, $B$, $C$ corresponding to sides $a$, $b$, $c$, vectors $\overrightarrow{m}=(\sin C, \cos C)$, $\overrightarrow{n}=(2\sin A - \cos B, -\sin B)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.  
$(1)$ Find the value of angle $C$.  
$(2)$ If $a=2$, find the range of values for the perimeter of $\triangle ABC$.

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