
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
Sixteen congruent disks are placed on a circle $C$ of radius 2 in such a way that the sixteen disks cover $C$, no two of the disks overlap, and each disk is tangent to its two neighbors. Calculate the sum of the areas of these sixteen disks. Express your answer in the form $\pi(a - b\sqrt{c})$, where $a, b, c$ are positive integers with $c$ not divisible by the square of any prime, and find $a+b+c$.

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