
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 $\begin{cases} & x=1+\cos \varphi \\ & y=\sin \varphi \\ \end{cases}$, 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.

  (I) Find the polar equation of the semicircle $C$;

  (II) The polar equation of a line is $\rho (\sin \theta +\sqrt{3}\cos \theta )=5\sqrt{3}$, the ray $OM$: $\theta =\frac{\pi }{3}$ intersects the semicircle $C$ at points $O$, $P$, and intersects the line at point $Q$. Find the length of the segment $PQ$.

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