
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
Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle  $C$  with radius  $30$ . Let  $K$  be the area of the region inside circle  $C$  and outside of the six circles in the ring. Find  $\lfloor K \rfloor$ .

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