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.