
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
An equilateral triangle is inscribed in a circle of radius $r$. Then, a circle is inscribed in that equilateral triangle, and then another equilateral triangle is inscribed in this latter circle, and so on. If $T_n$ is the sum of the areas of the first $n$ circles so inscribed, then, as $n$ grows beyond all bounds, $T_n$ approaches what value in terms of $\pi r^2$?
A) $\frac{12\pi r^2}{11}$
B) $\frac{\pi r^2}{2}$
C) $\frac{\pi r^2}{3}$
D) $\frac{3\pi r^2}{2}$
E) $\frac{\pi r^2}{4}$

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