
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 table. It rolls on the table surface without slipping, with the meeting point of the cone's base and table tracing a circular arc centered at the vertex touching the table. The cone returns to its initial position after completing $15$ full rotations. Determine the ratio $h/r$, expressed in the form $m\sqrt{n}$, where $m$ and $n$ are integers, and $n$ is not divisible by the square of any prime. Calculate $m + n$.

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