
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 horizontal surface. The cone completes $20$ full rotations before returning to its starting orientation. The point where the cone's base meets the surface traces a circular arc centered at the vertex of the cone. Find the ratio $h/r$ in the form $p\sqrt{q}$, where $p$ and $q$ are integers, and $q$ is not divisible by the square of any prime. Determine $p + q$.

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