
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 square has a large circle inscribed within it, and then a triangle is inscribed in that circle, followed by another circle inscribed in that triangle, thus forming a repeating sequence. If $m$ is the side length of the initial square and $S_n$ is the sum of the areas of the first $n$ circles, determine the value to which $S_n$ approaches as $n \to \infty$.
A) $\frac{\pi m^2}{4}$
B) $\frac{\pi m^2}{3}$
C) $\frac{\pi m^2}{4 - \sqrt{3}}$
D) $\frac{\pi m^2}{2}$
E) $\frac{3\pi m^2}{4}$

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