
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]


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In the rectangular coordinate system $(xOy)$, the parametric equations of circle $C$ are given by $\begin{cases} x=\cos \phi \ y=1+\sin \phi \end{cases}$ (where $\phi$ is the parameter). Establish a polar coordinate system with $O$ as the pole and the non-negative half of the $x$-axis as the polar axis.

(I) Find the polar equation of curve $C$;
(II) Suppose the polar equation of line $l$ is $\rho \sin (\theta + \frac{\pi }{3})=2$. Let $P$ be the intersection point of ray $OM$: $\theta = \frac{\pi }{6}$ with circle $C$, and $Q$ be the intersection point of the same ray with line $l$. Find the length of line segment $PQ$.

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