
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
Given a circle with area $A_1$ is contained in the interior of a larger circle with area $A_1 + A_2$, and the radius of the larger circle is $2\sqrt{3}$, and if $A_1, A_2, (A_1 + A_2 + A_3)$ form an arithmetic progression where $A_3$ is the area of an equilateral triangle with side length $s$, find the radius $r$ of the smaller circle.

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