2009 AIME I Problem 12

In right \triangle A B C with hypotenuse \overline{A B}, A C=12, B C=35, and \overline{C D} is the altitude to \overline{A B}. Let \omega be the circle having \overline{C D} as a diameter. Let I be a point outside \triangle A B C such that \overline{A I} and \overline{B I} are both tangent to circle \omega. The ratio of the perimeter of \triangle A B I to the length A B can be expressed in the form \frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.