Triangle A B C has side lengths A B=7, B C=8, and C A=9. Circle \omega_{1} passes through B and is tangent to line A C at A. Circle \omega_{2} passes through C and is tangent to line A B at A. Let K be the intersection of circles \omega_{1} and \omega_{2} not equal to A. Then A K=\frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.