AT-OEMO-FED1 2019 Problem 2

Let $ABC$ be a triangle and $I$ its incenter. The circle passing through $A, C$ and $I$ intersect the line $BC$ for second time at point $X$ . The circle passing through $B, C$ and $I$ intersects the line $AC$ for second time at point $Y$ . Show that the segments $AY$ and $BX$ have equal length.