
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 length $2m$ and width $m$, then a rectangle is inscribed in that circle, then a circle is inscribed in the latter rectangle, and so on. If $S_n$ is the sum of the areas of the first $n$ inscribed circles as $n$ grows, find the limit of $S_n$ as $n$ increases indefinitely.
A) $\frac{\pi m^2}{4}$  
B) $\frac{3\pi m^2}{8}$  
C) $\frac{\pi m^2}{3}$  
D) $\frac{\pi m^2}{2}$  
E) $\frac{\pi m^2}{8}$

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