
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 \(\boldsymbol{m}=(\sin A, \cos A)\) and \(\boldsymbol{n}=(\cos B, \sin B)\) with their dot product \(\boldsymbol{m} \cdot \boldsymbol{n}=\sin 2 C\), and \(A\), \(B\), and \(C\) are the angles opposite the sides \(a\), \(b\), and \(c\) of triangle \(\triangle ABC\):
(1) Find the measure of \(C\).
(2) If \(\sin A\), \(\sin C\), and \(\sin B\) form a geometric sequence, and \(\overrightarrow{CA} \cdot \overrightarrow{CB}=18\), find the value of \(c\).

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