
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
Imagine a regular a  $2015$ -gon with edge length  $2$ . At each vertex, draw a unit circle centered at that vertex and color the circle’s circumference orange. Now, another unit circle  $S$  is placed inside the polygon such that it is externally tangent to two adjacent circles centered at the vertices. This circle  $S$  is allowed to roll freely in the interior of the polygon as long as it remains externally tangent to the vertex circles. As it rolls,  $S$  turns the color of any point it touches into black. After it rolls completely around the interior of the polygon, the total length of the black lengths can be expressed in the form  $\tfrac{p\pi}{q}$  for positive integers  $p, q$  satisfying  $\gcd(p, q) = 1$ . What is  $p + q$ ?


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