SHARYGIN 2022 Problem 20

Let $O$ , $I$ be the circumcenter and the incenter of $\triangle ABC$ ; $R$ , $r$ be the circumradius and the inradius; $D$ be the touching point of the incircle with $BC$ ; and $N$ be an arbitrary point of segment $ID$ . The perpendicular to $ID$ at $N$ meets the circumcircle of $ABC$ at points $X$ and $Y$ . Let $O_{1}$ be the circumcircle of $\triangle XIY$ .

Find the product $OO_{1}\cdot IN$ .