A rigid cone rolls on a very rough, horizontal surface (rough enough to prevent it from sliding) and its center of mass follows a circular trajectory.

Suppose the angular velocity of the center of mass of the cone is \(\Omega\), and the angular velocity at which the cone rotates as seen by someone in the reference frame of the center of mass is \(\omega\).

What is the ratio of \(\Omega\) to \(\omega\)?

**Note:** The angle between the cone's base and slant height is \(\beta\)

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