
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 $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. Given that $c=3$ and $\sin (C-\frac{π}{6})\cdot \cos C=\frac{1}{4}$, find:<br/>
$(1)$ The measure of angle $C$;<br/>
$(2)$ If vectors $\overrightarrow{m}=\left(1,\sin A\right)$ and $\overrightarrow{n}=\left(2,\sin B\right)$ are collinear, find the perimeter of $\triangle ABC$.

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