A disk of mass $M$, radius $R$, and height $h$ is rotating about its center without friction at an angular speed $\omega_1$. A small rod points upward from the center of the disk. Two smaller disks are dropped one after another onto the large disk, causing the system to reach a new speed of $\omega_2$, and then a final speed of $\omega_3$, as shown above. The two disks have zero angular velocity, and each has dimensions that are half of the previous disk (e.g., the third disk is half the size of the second disk, which is half the size of the first disk).

What is the final angular speed $\omega_3$ in terms of $\omega_1$?

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