
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 base radius $r$ and height $h$. The cone rests sideways on a flat table. As it rolls without slipping along the table, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex of the cone meets the table. The cone returns to its initial position after making $19$ complete rotations. Write the ratio $h/r$ in the form $a\sqrt{b}$, where $a$ and $b$ are integers and $b$ is not divisible by the square of any prime. Find $a + b$.

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