
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 vectors $\overrightarrow{m}=({\cos A, \sin B})$, $\overrightarrow{n}=({\cos B, -\sin A})$, and $\overrightarrow{m} \cdot \overrightarrow{n} = \cos 2C$, where $A$, $B$, and $C$ are the interior angles of $\triangle ABC$, and $a$, $b$, $c$ are the opposite sides of angles $A$, $B$, $C$.  
$(1)$ Find $C$;  
$(2)$ If $2c=a+b$ and $ab\cos C=18$, find $c$.

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