In \triangle A B C, B C=23, C A=27, and A B=30. Points V and W are on \overline{A C} with V on \overline{A W}, points X and Y are on \overline{B C} with X on \overline{C Y}, and points Z and U are on \overline{A B} with Z on \overline{B U}. In addition, the points are positioned so that \overline{U V}\|\overline{B C}, \overline{W X}\| \overline{A B}, and \overline{Y Z} \| \overline{C A}. Right angle folds are then made along \overline{U V}, \overline{W X}, and \overline{Y Z}. The resulting figure is placed on a level floor to make a table with triangular legs. Let h be the maximum possible height of a table constructed from \triangle A B C whose top is parallel to the floor. Then h can be written in the form \frac{k \sqrt{m}}{n}, where k and n are relatively prime positive integers and m is a positive integer that is not divisible by the square of any prime. Find k+m+n.
