A string is wound around a cone with base radius 1 cm and slant height 10 cm. The string is wrapped in a helical manner, as if a strand of lights decorating a Christmas tree, except that it is pulled taut around the cone so as to have no local "wiggles", thus making what is known mathematically as a geodesic curve on the surface of the cone.

What is the maximum length of such a spiral around the cone before it starts to intersect itself?