
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 given that the center of a circle is $C(3, \frac{\pi}{6})$ and the radius $r=1$.
(1) Find the polar equation of the circle;
(2) If the line $\begin{cases} x=-1+\frac{\sqrt{3}}{2}t \\ y=\frac{1}{2}t \end{cases}$ (where $t$ is a parameter) intersects the circle at points A and B, find the distance between the midpoint C of the segment AB and the point P(-1, 0).

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