TxO Math Bowl 2024 - Individuals B - Problem 5

In \triangle ABC, let points D and E be on AC such that BD and BE trisect \angle ABC and D is between A and E. Let F be a point on the extension of CA such that \angle FBA = \frac{1}{3}\left(180^\circ - \angle ABC\right). Given that BC = 6, EC = 3, DE = 2, FD = 4\sqrt{3}, compute the length of BF.