
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
Construct a rectangle with the largest area inside the ellipse \(b^{2} x^{2} + a^{2} y^{2} = a^{2} b^{2}\) (where \(a > b\)) with sides parallel to the major axes of the ellipse. Inside this rectangle, inscribe an ellipse whose major axes are equal to the sides of the rectangle. Construct a rectangle with the largest area inside this new ellipse, and repeat this process indefinitely. What is the sum of the areas of the infinitely many rectangles and ellipses thus created?

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