
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) Given that point $P$ lies on an ellipse with the coordinate axes as its axes of symmetry, the distances from point $P$ to the two foci are $\frac{4\sqrt{5}}{3}$ and $\frac{2\sqrt{5}}{3}$, respectively. A line perpendicular to the major axis and passing through point $P$ intersects one of the foci. Find the equation of the ellipse.

(2) The focal distance of a hyperbola is $\sqrt{5}$ times the length of its real axis, and one of its vertices has coordinates $(0,2)$. Find the equation of the hyperbola.

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