
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
Given an ellipse centered at the origin with its foci on the $x$-axis and an eccentricity of $\frac{1}{2}$, and a point $P$ on the ellipse has a sum of distances to the two foci equal to $8$,  
$(1)$ Find the equation of the ellipse;  
$(2)$ Find the equation of the hyperbola that shares the same foci with the above ellipse and has an asymptote of $y= \sqrt {3}x$.

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