
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 equations of circle $C$ are given by $\begin{cases}x=1+\cos φ \\\\ y=\sin φ\end{cases}$ ($φ$ 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.

(1) Find the Cartesian equation and the polar equation of circle $C$.

(2) The polar equation of line $l$ is $2ρ\sin \left(θ+ \dfrac{π}{3}\right)=6 \sqrt{3}$. Let $OM$ be the ray with $θ= \dfrac{π}{6}$, intersecting circle $C$ at points $O$ and $P$, and intersecting line $l$ at point $Q$. Find the length of line segment $PQ$.

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