
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 the polar equation of curve $C$ is $\rho=4\cos \theta$, and a Cartesian coordinate system is established with the pole as the origin and the positive half-axis of the $x$-axis as the polar axis. Suppose the parametric equation of line $l$ is $\begin{cases} x=5+ \frac{\sqrt{3}}{2}t \\ y=\frac{1}{2}t \end{cases}$ (where $t$ is the parameter).

$(1)$ Find the Cartesian equation of curve $C$ and the general equation of line $l$;  
$(2)$ Suppose curve $C$ intersects line $l$ at points $P$ and $Q$, and a rectangle inscribed in curve $C$ is formed with $PQ$ as one of its sides. Find the area of this rectangle.

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