
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 \( h \) lies on its side on a flat surface. As it rolls without slipping along the table, the point where the base meets the surface traces a circular arc centered at the cone’s vertex. This arc constitutes a full circle once the cone completes \( 20 \) rotations. Calculate the ratio \( h/r \), which can be expressed in the form \( m\sqrt{n} \) where \( m \) and \( n \) are positive integers, and \( n \) is not divisible by the square of any prime. Find \( m + n \).

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