
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 $A$, $B$, and $C$ are the angles opposite to sides $a$, $b$, and $c$ of $\triangle ABC$ respectively, and vectors $\overrightarrow{m}=(\sin A,\sin B)$, $\overrightarrow{n}=(\cos B,\cos A)$, and $\overrightarrow{m}\cdot \overrightarrow{n}=\sin 2C$.  
$(1)$ Find the size of angle $C$;  
$(2)$ If $\sin A$, $\sin C$, $\sin B$ form an arithmetic sequence, and $\overrightarrow{CA}\cdot( \overrightarrow{AB}- \overrightarrow{AC})=18$, find the length of side $c$.

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