IMO 2026 Problem 2

Let $ABC$ be a triangle and let points $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Let points $K$ and $L$ be chosen strictly inside triangles $BMC$ and $BNC$, respectively, such that $K$ lies strictly inside triangle $ABL$ and $L$ lies strictly inside triangle $AKC$. Suppose that

$$\angle KBA = \angle ACL, \quad \angle LBK = \angle LNC, \quad \text{and} \quad \angle LCK = \angle BMK.$$

Let $O$ be the circumcentre of triangle $AKL$. Prove that $OM = ON$.