
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 smaller circle is contained inside a larger circle. The radius of the larger circle is $5$. The areas of the smaller and larger circles form an arithmetic progression with the sum of their areas. If the areas of the larger circle, the sum of both areas, and the smaller circle's area are in that order to form this arithmetic sequence, find the radius of the smaller circle.
A) $2\sqrt{2}$
B) $5$
C) $5\sqrt{2}$
D) $10$
E) $7$

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