Find the largest r such that 4 balls each of radius r can be packed into a regular tetrahedron with side length 1. In a packing, each ball lies outside every other ball, and every ball lies inside the boundaries of the tetrahedron. If r can be expressed in the form \frac{\sqrt{a}+b}{c} where a, b, c are integers such that \operatorname{gcd}(b, c)=1, what is a+b+c?