
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
A rectangle with side lengths $a$ and $2a$ has an ellipse inscribed in it where the major radius equals $a$ and the minor radius equals $a/2$. Subsequently, an ellipse is inscribed inside the largest possible rectangle fitting inside this initial ellipse, and the process continues infinitely. Assuming $S_n$ is the sum of the areas of the first $n$ ellipses, what does $S_n$ approach as $n$ grows beyond all bounds?
A) $\frac{\pi a^2}{2}$
B) $\pi a^2$
C) $2\pi a^2$
D) $\frac{\pi a^2}{4}$
E) $\frac{3\pi a^2}{4}$

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