DE-BUNDESWETTBEWERB 2011 Problem 8

Let $ABCD$ be a tetrahedron that is not degenerate and not necessarily regular, where sides $AD$ and $BC$ have the same length $a$ , sides $BD$ and $AC$ have the same length $b$ , side $AB$ has length $c_1$ and the side $CD$ has length $c_2$ . There is a point $P$ for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities $a, b, c_1$ and $c_2$ .