Given a circle with radius $r$ and a segment $AB$ parallel to the plane of the circle, at a distance $r$ from the plane. The orthogonal projection of the segment $AB$ onto the plane of the circle is a diameter of the circle. From every interior point of the segment $AB$, we draw two half-lines perpendicular to $AB$, each of which intersects the circle. Determine the volume of the solid bounded by the circle and the half-lines.

The problem involves a circle with radius $r$ and a segment $AB$ that is parallel to the plane of the circle and is at a distance $r$ from the plane. The orthogonal projection of the segment $AB$ onto the plane of the circle is a diameter of the circle. From every interior point of the segment $AB$, we draw two half-lines perpendicular to $AB$, each of which intersects the circle. We need to determine the volume of the solid bounded by the circle and the half-lines.