
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 on its side on a flat table. As the cone rolls on the surface of the table it traces a circular arc with the tip acting as a pivot, and it completes its initial positioning after making \( 10 \) complete rotations. Find the value of \( \frac{h}{r} \) and express it in the simplest form of \( m\sqrt{n} \), where \( m \) and \( n \) are positive integers and \( n \) is not divisible by the square of any prime. What is the sum \( m + n \)?

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