
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
Eight congruent disks are placed on a circle $C$ of radius 1 in such a way that the eight disks cover $C$, no two of the disks overlap, and so that each of the eight disks is tangent to its two neighbors. The sum of the areas of the eight disks can be written in the form $\pi(a-b\sqrt{c})$, where $a, b, c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$.

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