
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 $\overrightarrow{m}=(\sin A,\cos A)$ and $\overrightarrow{n}=(\cos B,\sin B)$, with $\overrightarrow{m}\cdot \overrightarrow{n}=\sin 2C$, where $A$, $B$, and $C$ are the angles opposite to sides $a$, $b$, and $c$ of $\triangle ABC$, respectively.

1. Find the measure of angle $C$.
2. If $\sin A$, $\sin C$, and $\sin B$ form a geometric sequence, and $\overrightarrow{CA}\cdot (\overrightarrow{AB}-\overrightarrow{AC})=18$, find the value of $c$.

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