
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
A right circular cone with base radius \(r\) and height \(h\) rolls on its side on a flat surface. It completes \(20\) full rotations before returning to its original position. The point where the cone's base meets the surface traces a circular path centered at the cone's vertex. Find the value of \(h/r\) in the simplest form \(a\sqrt{b}\), where \(a\) and \(b\) are integers with \(b\) not divisible by the square of any prime, and provide \(a + b\).

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