
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 total distance to the two foci of 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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