
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
Rotate the ellipse \( b^{2} x^{2} + a^{2} y^{2} = a^{2} b^{2} \) around its minor axis. In the resulting ellipsoid, inscribe the largest volume right circular cylinder. (The cylinder's axis is the rotation axis of the ellipsoid, and the bases of the cylinder lie on the ellipsoid.) In this cylinder, inscribe a rotating ellipsoid that touches the lateral surface, base, and top of the cylinder. In this newly obtained ellipsoid, construct the largest volume right circular cylinder inscribed within it. What is the sum of the volumes of these infinite right circular cylinders and ellipsoids? What is the ratio of these two sums?

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