
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
Selected Exercise $4-4$: Polar Coordinates and Parametric Equations

In the rectangular coordinate system $xOy$, curve $B$: $({x}^{2}+{y}^{2}=1)$ undergoes a scaling transformation $\begin{cases}{x}^{{{{'}}}}=3x \\\\ {y}^{{{{'}}}}=y\\end{cases}$, resulting in curve $C$.

$(1)$ Find the rectangular coordinate equation of curve $C$;

$(2)$ Find a point $D$ on curve $C$ such that its distance to line $l$: $x+4y-8=0$ is minimized, and find the rectangular coordinates of point $D$.

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