
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 for the following hyperbolas:
(1) A hyperbola that shares its foci with the ellipse $\frac{x^2}{16} + \frac{y^2}{25} = 1$ and passes through the point $(-2, \sqrt{10})$.
(2) A hyperbola whose asymptotes are $x \pm 2y = 0$ and that passes through the point $(2,2)$.

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