
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 has base radius $r$ and height $h$. The cone lies horizontally on a table. As the cone rolls without slipping, the point where the cone's base meets the table traces a circular arc centered at its vertex. The cone returns to its starting position after completing $20$ full rotations. The value of $h/r$ can be written 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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