Let \triangle A B C be a triangle, and let l be the line passing through its incenter and centroid. Assume that B and C lie on the same side of l, and that the distance from B to l is twice the distance from C to l. Suppose also that the length B A is twice that of C A. If \triangle A B C has integer side lengths and is as small as possible, what is A B^{2}+B C^{2}+C A^{2}?