
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) Find the standard equation of an ellipse whose right focus is at the coordinate $(2, 0)$ and that passes through the point $(-2, -\sqrt{2})$;
(2) Find the equation of a hyperbola with a common focus to the ellipse $\frac{x^2}{24} + \frac{y^2}{49} = 1$, and with a common asymptote to the hyperbola $\frac{x^2}{36} - \frac{y^2}{64} = 1$.

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