
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 $xOy$, the parametric equations of curve $C_{1}$ are $\left\{\begin{array}{l}{x=co{s}^{k}t,}\\{y=si{n}^{k}t}\end{array}\right.$ (where $t$ is the parameter). Establish a polar coordinate system with the origin as the pole and the positive x-axis as the polar axis. The polar coordinate equation of curve $C_{2}$ is $4\rho \cos \theta -16\rho \sin \theta +3=0$.  
$(1)$ When $k=1$, what curve is $C_{1}$?  
$(2)$ When $k=4$, find the Cartesian coordinates of the intersection points of $C_{1}$ and $C_{2}$.

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