PUMaC 2019 Geometry A Problem 4

Let B C=6, B X=3, C X=5, and let F be the midpoint of B C. Let A X \perp B C and A F=\sqrt{247}. If A C is of the form \sqrt{b} and A B is of the form \sqrt{c} where b and c are nonnegative integers, find 2 c+3 b.