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In a triangle $A B C, I$ is the incenter and $D, E, F$ are the points of tangency of the incircle with $B C, C A, A B$, respectively. The bisector of angle $B I C$ meets $B C$ at $M$, and the line $A M$ intersects $E F$ at $P$. Prove that $D P$ bisects the angle $F D E$.
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