
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 a base radius $r$ and height $h$. The cone is laid on its side on a flat table. As it rolls on the surface without slipping, the point where the cone's base meets the table traces a circular arc centered at the vertex touching the table. The cone first returns to its original position after making $23$ complete rotations. Express the ratio $h/r$ in the form $p\sqrt{q}$, where $p$ and $q$ are positive integers with $q$ not divisible by the square of any prime. Find $p + q$.

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