CMO 2023 Problem 3

An acute triangle is a triangle that has all angles less than $90^{\circ}$ ( $90^{\circ}$ is a Right Angle). Let $A B C$ be an acute triangle with altitudes $A D, B E$, and $C F$ meeting at $H$. The circle passing through points $D, E$, and $F$ meets $A D, B E$, and $C F$ again at $X, Y$, and $Z$ respectively. Prove the following inequality:

$$\frac{A H}{D X}+\frac{B H}{E Y}+\frac{C H}{F Z} \geq 3$$