
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 parametric equation of the semicircle $(C)$ is $\begin{cases}x=1+\cos \left(\phi\right) \\ y=\sin \left(\phi\right)\end{cases} (\phi$ is the parameter, $0\leqslant \phi\leqslant \pi)$. 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 polar equation of $(C)$;

(2) The polar equation of the line $(l)$ is $\rho(\sin \theta+ \sqrt{3}\cos \theta)=5 \sqrt{3} $. The intersections of the ray $OM$: $\theta= \dfrac{\pi}{3} $ with the semicircle $(C)$ are $O$ and $P$, and with the line $(l)$ is $Q$. Find the length of the segment $PQ$.

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