We have a triangle $\triangle ABC$ in which angles $B$ and $C$ are greater than $45^\circ$. We construct two rectangular isosceles triangles $\triangle BAN$ and $\triangle CAM$ out of $\triangle ABC$ with angles $\angle CAM = \angle BAN = 90^\circ$. Then we draw a rectangular isosceles triangle in the triangle $ABC$ with angle $P$ equal to $90$ degrees. Prove that $\triangle MPN$ is rectangular isosceles.
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