
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 1 such that these sixteen disks cover \(C\), no two of the disks overlap, and each disk is tangent to its two neighbors. Find the sum of the areas of the sixteen disks, which can be written in the form \(\pi(a-b\sqrt{c})\), where \(a\), \(b\), and \(c\) are positive integers, and \(c\) is not divisible by the square of any prime. Provide the value of \(a+b+c\).

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