Let $O, I$ be the circumcenter and the incenter of triangle $ABC$, $P$ be an arbitrary point on segment $OI$, $P_A$, $P_B$, and $P_C$ be the second common points of lines $PA$, $PB$, and $PC$ with the circumcircle of triangle $ABC$. Prove that the bisectors of angles $BP_AC$, $CP_BA$, and $AP_CB$ concur at a point lying on $OI$.