
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 $h$ rests on its side on a horizontal surface. As it rolls on the surface without slipping, the point where the cone’s base meets the surface traces a circular arc centered at the vertex touching the surface. The cone returns to its original position after making $10$ complete rotations. Find the ratio $h/r$ in its simplest form $m\sqrt{n}$, where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime, and calculate $m+n$.

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