APMO 2026 Problem 4

Let $ABCD$ be a quadrilateral with an incircle $\omega$ of centre $I$. The diagonals $AC$ and $BD$ intersect at $E$. Let $J$ be the incentre of triangle $ABD$. The extension of the ray $EJ$ intersects $\omega$ at $P$. Prove that $PI \perp BD$.