
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 a base radius $r$ and height $h$ lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base connects with the table traces a circular arc centered at the vertex of the cone. The cone first returns to its original position after making $20$ complete rotations. The value of $h/r$ in simplest form can be expressed as $\lambda\sqrt{k}$, where $\lambda$ and $k$ are positive integers, and $k$ is not divisible by the square of any prime. Find $\lambda + k$.

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