
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
Given that the pole of a polar coordinate system is at the origin of a Cartesian coordinate system, and the polar axis coincides with the non-negative x-axis, the parametric equations of line $l$ are: $\begin{cases} x=-1+\frac{\sqrt{3}}{2}t \\ y=\frac{1}{2}t \end{cases} (t \text{ is the parameter})$, and the polar equation of curve $C$ is: $\rho=4\cos \theta$.

(1) Write the Cartesian equation of curve $C$ and the normal equation of line $l$;
(2) Suppose line $l$ intersects curve $C$ at points $P$ and $Q$, find the value of $|PQ|$.

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