
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, let circle $C$ pass through point $P(\sqrt{3}, \frac{\pi}{6})$, and the center of the circle is the intersection point of line $\rho \sin(\frac{\pi}{3} - \theta) = \frac{\sqrt{3}}{2}$ and the polar axis.
(1) Find the radius of circle $C$;
(2) Find the polar coordinate equation of circle $C$.

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