
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 angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively, and $\sin A \cdot \cos B + \sin B \cdot \cos A = \sin 2C$. If $a$, $b$, $c$ form an arithmetic sequence and $\vec{CA} \cdot \vec{CB} = 18$, calculate the length of side $c$.

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