
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 Cartesian equation of the semicircle $(C)$ is $((x-1)^2+y^2=1 (0 \leqslant y \leqslant 1))$. 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 $ρ(sin θ+ \sqrt{3}cos θ)=5 \sqrt{3}$. The ray $OM: θ= \frac{π}{3}$ intersects the semicircle $(C)$ at points $O$ and $P$, and intersects the line $(l)$ at point $Q$. Find the length of the line segment $PQ$.

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