An Insect On A Disc

A uniform, circular disc of mass $M$ lying on a smooth, horizontal surface is free to rotate about a vertical axis $O$ passing through its rim. An insect of mass $m$ is at point $A$ such that line $OA$ is the diameter of the disc. The insect describes a complete circle relative to the disc and returns to the starting point $A.$

If the angle moved by the disc relative to the ground is given by $\pi\left(1-\sqrt{\frac{xM}{yM+zm}}\right),$ then the value of $x+y+z$ is $\text{\_\_\_\_\_\_\_\_\_\_}.$