
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
Fifteen congruent disks are placed on a circle \(C\) of radius 1 in such a way that the fifteen disks cover \(C\), no two of the disks overlap, and each of the fifteen disks is tangent to its two neighbors. Find the sum of the areas of the fifteen disks, which can be written 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. Determine \(a+b+c\).

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