
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 semi-circle $(C)$ is given by $\begin{cases} & x=1+\cos \varphi , \ & y=\sin \varphi \ \end{cases}(\varphi)$ is the parameter, $0\leqslant \varphi\leqslant \pi).$ 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 coordinate equation of $(C)$;

(2) The polar coordinate equation of the line $(l)$ is $\rho (\sin \theta +\sqrt{3}\cos \theta )=5\sqrt{3}$, the ray $OM$: $\theta =\dfrac{\pi }{3}$ intersects the semi-circle $(C)$ at points $O$, $P$, and the line $(l)$ at point $Q$. Calculate the length of the line segment $PQ$.

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