
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
Find the standard equations of the conic sections that fulfill the following conditions:  
(1) An ellipse whose major axis is twice the length of its minor axis and passes through point P(2,0);  
(2) A hyperbola with foci on the y-axis, semi-major axis \(a=2\sqrt{5}\), and passes through point A(2,5);  
(3) A parabola with the vertex at the origin, axes of symmetry aligned with the coordinate axes, and passes through point P(1,-2).

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