In rectangle ABCD, AB = 6 and BC = 3. Point E between and B and C, and point F between E and C are such that BE=EF=FC. Segments AE and AF intersect BD at P and Q, respectively. The ratio BP : PQ : QD can be written as r : s : t, where the greatest common factor of r, s, and t is 1. What is r + s +t?