
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 $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, a semicircle $C$ is described by the polar equation $\rho=2\cos \theta$, $\theta\in[0, \frac{\pi}{2}]$.

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

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