2023 IOQM/Problem 14

Let ABC be a triangle in the xy plane, where B is at the origin (0, 0). Let BC be produced to D such that BC : CD = 1 : 1, CA be produced to E such that CA : AE = 1 : 2 and AB be produced to F such that AB : BF = 1 : 3. Let G(32, 24) be the centroid of the triangle ABC and K be the centroid of the triangle DEF. Find the length GK.