The shaded region in the figure is called an arbelos, formed by three mutually tangent semicircles. The incircle is the largest circle that can fit inside the arbelos, and it is tangent to all three semicircles. If the largest semicircle has a diameter $AB$ of $1$, what should be the diameter $x$ of one of the smaller semicircles to maximize the radius of the incircle $r$? Note that the radius of the incircle can be expressed in terms of the diameters of the two smaller semicircles as follows:

$r=\frac{xy}{2(x+y_{})}$