Two identical small beads of mass \(m \text{ kg}\) each are attached to a ring (of mass \( m\text{ kg}\)), and the total system is kept on a long horizontal frictionless table. Beads are free to move around the ring and there is no friction between the ring and the beads.

Both the beads are given an initial velocity of \(v \text{ m/s}\) and the ring is initially at rest.

Let the magnitude of velocity of the ring at the instant when the two beads are about to collide be \(x\text{ m/s},\) and let the magnitude of velocity of the lower bead in the diagram be \(y\text{ m/s}\).

Find the value of \(\left(\dfrac yx\right)^2\).

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