The incircle of acute triangle A B C touches B C, A C, and A B at points D, E, and F, respectively. Let P be the second intersection of line A D and the incircle. The line through P tangent to the incircle intersects A B and A C at points M and N, respectively. Given that \overline{A B}=8, \overline{A C}=10, and \overline{A N}=4, let \overline{A M}=\frac{a}{b} where a and b are positive coprime integers. What is a+b?