
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
Elective 4-4: Coordinate System and Parametric Equations

In the Cartesian coordinate system $xOy$, the curve $C$ is given by the parametric equations:

\[
\begin{cases}
x=1+ \sqrt{3}\cos \varphi \\
y= \sqrt{3}\sin \varphi
\end{cases}
\]
where $\varphi$ is the parameter, $0 \leqslant \varphi \leqslant \pi$. A polar coordinate system is established 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 $2\rho\sin (\theta+ \frac{\pi}{3})+3 \sqrt{3}=0$, and line $l_{2}: \theta= \frac{\pi}{3} (\rho \in \mathbb{R})$ 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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