SHARYGIN 2019 Problem 13

Let $ABC$ be an acute-angled triangle with altitude $AT = h$ . The line passing through its circumcenter $O$ and incenter $I$ meets the sides $AB$ and $AC$ at points $F$ and $N$ , respectively. It is known that $BFNC$ is a cyclic quadrilateral. Find the sum of the distances from the orthocenter of $ABC$ to its vertices.