
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 rectangular coordinate system $xOy$, the parametric equation of the curve $C$ is given by $\begin{cases} x=1+\cos\theta \\ y=\sin\theta \end{cases}$ ($\theta$ is the parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis.
(1) Find the polar equation of the curve $C$.
(2) The polar equation of the line $l$ is $\rho(\sqrt{3}\sin\theta + \cos\theta) = 4$. The intersection points of the ray $OM: \theta = \frac{\pi}{3}$ with the curve $C$ are $O$ and $P$, and its intersection point with the line $l$ is $Q$. Find the length of the line segment $PQ$.

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