Let \(ABC\) be an acute triangle with altitude \(\overline{AH}\), and let \(P\) be a variable point such that the angle bisectors \(k\) and \(\ell\) of \(\angle PBC\) and \(\angle PCB\), respectively, meet on \(\overline{AH}\). Let \(k\) meet \(\overline{AC}\) at \(E\), \(\ell\) meet \(\overline{AB}\) at \(F\), and \(\overline{EF}\) meet \(\overline{AH}\) at \(Q\). Prove that as \(P\) varies, line \(PQ\) passes through a fixed point.