
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 with area \(A_1\) is inscribed within a larger circle whose total area is \(A_1 + A_2\). If the radius of the larger circle is \(8\), and \(A_1, A_2,\) and \(A_1 + A_2\) form an arithmetic progression, find the radius of the smaller circle.
A) $\frac{5\sqrt{3}}{3}$  
B) $\frac{7\sqrt{3}}{3}$  
C) $\frac{8\sqrt{3}}{3}$  
D) $\frac{9\sqrt{3}}{3}$  
E) $\frac{10\sqrt{3}}{3}$

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