Let $\triangle ABC$ be a triangle with incentre $I$. Points $E$ and $F$ are the tangency points of the incircle and the sides $AC$ and $AB$, respectively. Suppose that the lines $BI$ and $CI$ intersect the line $EF$ at $Y$ and $Z$, respectively. Let $M$ denote the midpoint of the segment $BC$, and $N$ denote the midpoint of the segment $YZ$. Prove that $AI \parallel MN$.