Four, (15 points) As shown in the figure, in $\triangle A B C$, $\angle B=90^{\circ}, M$ is a point on $A B$ such that $A M = B C, N$ is a point on $B C$ such that $C N = B M$. Connect $A N, C M$ intersecting at point $P$. Try to find the degree of $\angle A P M$ and write out your reasoning and proof process.