
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 Cartesian coordinate system $xOy$, a circle $C$ has a parametric equation 
$$\begin{cases}x = 2 - 2\cos \phi \\ y = 2\sin \phi\end{cases},$$ where $\phi \in \mathbb{R}$. Establish a polar coordinate system with the origin as the pole and the non-negative half-axis of the $x$-axis as the polar axis. 
$(I)$ Find the polar equation of circle $C$; 
$(II)$ The polar equations of line $l$ and ray $OM$ are $\rho\cos (\theta - \frac{\pi}{6}) = 3\sqrt{3}$ and $\theta = \frac{\pi}{3}$, respectively. If ray $OM$ intersects circle $C$ at points $O$ and $P$, and intersects line $l$ at point $Q$, find the value of $|PQ|$.

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