
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 the vectors $\boldsymbol{m}=(\sin A, \cos A)$, $\boldsymbol{n}=(\cos B, \sin B)$, $\boldsymbol{m} \cdot \boldsymbol{n}=\sin 2C$, and the angles $\angle A$, $\angle B$, and $\angle C$ correspond to the sides $a$, $b$, and $c$ of triangle $\triangle ABC$ respectively:
1. Find the measure of $\angle C$.
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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