
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 an ellipse with its foci on the $y$-axis, the length of the major axis is $10$, and the length of the minor axis is $8$. Let $F_1$ and $F_2$ be the left and right foci of the ellipse, respectively.  
$(1)$ Find the standard equation of the ellipse;  
$(2)$ Find the coordinates of the foci and the eccentricity of the ellipse;  
$(3)$ Find the standard equation of the hyperbola with the foci of the ellipse as its vertices and the vertices as its foci.

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