
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
Determine the standard equations of the hyperbolas that satisfy the following conditions:
(Ⅰ) The foci are on the y-axis, the focal distance is 16, and the eccentricity \( e = \frac{4}{3} \);
(Ⅱ) An equilateral hyperbola with one focus at \( F(-6, 0) \).

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