
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 $xOy$, the curve $C$ is given by the parametric equations: 
$$
\begin{cases} 
x=1+ \sqrt {3}\cos \phi \\
y= \sqrt {3}\sin \phi 
\end{cases}
$$
where $\phi$ is the parameter, $0\leqslant \phi\leqslant \pi$. Establish a polar coordinate system with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.

$(1)$ Find the polar equation of curve $C$;  
$(2)$ The polar equation of line $l_{1}$ is $2\rho\sin (\theta+ \dfrac {\pi}{3})+3 \sqrt {3}=0$, and line $l_{2}$: $\theta= \dfrac {\pi}{3}(\rho\in\mathbb{R})$ intersects curve $C$ at point $P$ and intersects line $l_{1}$ at point $Q$. Find the length of segment $PQ$.

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