
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 $h$ is placed upright. A small bead slides along the surface from the vertex to the base along the shortest path (a straight line on the cone's surface) as the cone rotates about its vertical axis, completing $10$ full rotations. The ratio $h/r$ can be written in the form $p\sqrt{q}$, where $p$ and $q$ are positive integers and $q$ is not divisible by the square of any prime. Find $p + q$.

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