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In a triangle $A B C, A E$ and $B F$ are altitudes and $H$ the orthocenter. The line symmetric to $A E$ with respect to the bisector of $\angle A$ and the line symmetric to $B F$ with respect to the bisector of $\angle B$ intersect at a point $O$. The lines $A E$ and $A O$ meet the circumcircle of $\triangle A B C$ again at $M$ and $N$, respectively. The lines $B C$ and $H N$ meet at $P, B C$ and $O M$ at $R$, and $H R$ and $O P$ at $S$. Prove that $A H S O$ is a parallelogram.
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