
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
(1) An ellipse with its foci on the y-axis has a vertex at A(2, 0), and its major axis is twice the length of its minor axis. Find the standard equation of the ellipse.  
(2) Given that one of the asymptotes of a hyperbola is $x + 2y = 0$, and it passes through the point (2, 2), find the standard equation of this hyperbola.

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