
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 circle is inscribed in a rectangle with sides \(a\) and \(b\) (where \(a \leq b\)), then a rectangle that maximally fits within that circle is inscribed, then a circle is inscribed in that rectangle, and so on. If \(S_n\) is the sum of the areas of the first \(n\) circles so inscribed, then, as \(n\) grows beyond all bounds, \(S_n\) approaches:
A) \(\frac{\pi a^2}{2}\)
B) \(\frac{3\pi a^2}{8}\)
C) \(\frac{\pi a^2}{3}\)
D) \(\frac{\pi a^2}{4}\)
E) \(\frac{\pi a^2}{8}\)

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