Floating beads in parabolic wires

A bead of mass \(m\) slides freely along a frictionless wire in the shape of the curve \(y = x^{2}\) rotates around the \(y\)-axis with a constant angular velocity \(\omega\). There is a constant downward gravitational acceleration \(g\).

Suppose the bead maintains a constant height above the ground so that it "floats." How many such heights are possible?

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