Let A B C be a triangle with A B=13, B C=14, and A C=15. Denote by \omega its incircle. A line \ell tangent to \omega intersects \overline{A B} and \overline{A C} at X and Y respectively. Suppose X Y=5. Compute the positive difference between the lengths of \overline{A X} and \overline{A Y}.