
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
In the polar coordinate system, it is known that the circle $C$ has its center at $C(3, \frac{\pi}{6})$ and radius $r=3$.  
$(1)$ Find the polar equation of circle $C$;  
$(2)$ If point $Q$ moves on circle $C$ and point $P$ is on the extension of line $OQ$, with $|OQ|:|QP|=3:2$, find the trajectory equation of the moving point $P$.

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