
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]


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Elective 4-4: Coordinate System and Parametric Equations

In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of the semicircle $C$ is $\rho =2\cos \theta ,\theta \in \left[ 0,\dfrac{\pi }{2} \right]$.

$(1)$ Find the parametric equation of $C$;

$(2)$ Let point $D$ be on $C$, and the tangent line of $C$ at $D$ is perpendicular to the line $l:y=\sqrt{3}x+2$. Based on the parametric equation you obtained in $(1)$, determine the coordinates of $D$.

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