Example 5 Given in $\square A B C D$, $E, F$ are points on sides $A B, A D$ respectively, and $E F$ intersects diagonal $A C$ at point $P$. If $\frac{A E}{E B}=\frac{a}{b}, \frac{A F}{F D}=\frac{m}{n}\left(a, b, m, n \in \mathbf{R}_{+}\right)$, then the value of $\frac{A P}{P C}$ is $(\quad$.
(A) $\frac{a m}{a m+b m}$
(B) $\frac{b n}{a n+b m}$
(C) $\frac{a m}{a m+a n+b m}$
(D) $\frac{b n}{a n+b m+b n}$
(2008, National Junior High School Mathematics Competition, Tianjin Preliminary)