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

$(1)$ Convert the polar equation of curve $C$ into a Cartesian coordinate equation;

$(2)$ Suppose the Cartesian coordinates of point $M$ are $(5,\sqrt{3})$, and the intersection points of line $l$ and curve $C$ are $A$ and $B$. Find the value of $|MA|\cdot |MB|$.

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