
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 square with side length $10$ units. Inside this circle, a new square is then inscribed, and then a new circle inside this square, continuing the process. If $S_n$ is the sum of the areas of the first $n$ concentric circles, and if the side length of each newly inscribed square is half of the square within which it is inscribed, what does $S_n$ approach as $n$ becomes very large?
A) $\frac{200\pi}{3}$
B) $50\pi$
C) $\frac{100\pi}{3}$
D) $100\pi$
E) $\frac{50\pi}{3}$

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