
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
Transform every point on the ellipse $x^{2}+ \frac {y^{2}}{4}=1$ by keeping the x-coordinate unchanged and changing the y-coordinate to half of its original value to obtain curve $C$.  
$(1)$ Write the parametric equation of curve $C$;  
$(2)$ Suppose point $D$ is on curve $C$, and the tangent line of $C$ at point $D$ is perpendicular to the line $l$: $y=x+2$, find the coordinates of $D$.

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