
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
Given a hyperbola with its foci on the $x$-axis, the length of its real axis is $10$, and the length of its imaginary axis is $8$. Let $F_1$ and $F_2$ be the left and right foci of the hyperbola, respectively.

$(1)$ Find the standard equation of the hyperbola.

$(2)$ Find the length of the imaginary axis and the eccentricity of the hyperbola.

$(3)$ Find the standard equation of the ellipse with the vertices of the hyperbola as its foci and the foci $F_1$, $F_2$ as its vertices.

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