Golden Spire

There are some peculiar facts known about the bacterium 'Jandelinka Fest', which inhabits the Jandal region. When a strain of this bacterium is exposed to air, it begins to spread uniformly in all directions so that the boundary of its population becomes an ideal sphere centered at the initial location of the strain. The radius of the sphere increases linearly with time.

Moreover, the bacteria cannot tolerate a certain aroma. If they sense cigarette smoke, they immediately die.

This property is often used by local priests. They decorate the temples with golden spires of fanciful shapes in the following manner. An ideally straight, thin, vertical stick is set up on a flat horizontal ground surface. Certain points on the stick are inoculated with bacterial strains (different strains at different points). Since colonies may be of different species, they may have different rates of propagation. After a week, the monks stop the spreading process by smoking the stick and then cover the resulting construction with golden dust. The spire is then mounted on the peak of the temple.

The spheres corresponding to different colonies may intersect, but the bacteria are completely indifferent to one another. A point is considered covered by a colony if and only if it is contained in any of the spheres (each growing independently). Furthermore, when the bacteria reach ground level (zero height), they immediately die.

You are required to find the total area of the spire's surface that is covered after the stick has been processed by bacteria, treating the stick as an ideal line. The monks need this to estimate the amount of gold required to finish the spire. Only the spherical parts of the surface should be counted in the calculation.