
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
Out of 8 circular disks with radius \( r \), 7 are fixed on a table such that their centers are at the vertices and center of a regular hexagon with side length \( 2r \), and the 8th disk touches one of the 6 outer disks. The 8th disk is rolled around the 7 fixed disks once without slipping, until it returns to its starting position. How many times does the 8th disk rotate about its own center during this process? (Provide an explanation for the answer.)

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