Let $ABC$ be a triangle and $E$ and $F$ lie on $AC$ and $AB$ such that $AE=AF$. $EF$ intersects $BC$ at $D$ and $(BDF)$ intersects $(CDE)$ at $X$. Let $O_1$ be the center of $(BDF)$ and $O_2$ be the center of $(CDE)$. Let $O$ be the center of $ABC$. Suppose that $XD$ intersects $(XO_1O_2)$ at $Z$. Show that $OZ\parallel BC$. (Proposed by Tan Rui Xuen and Yeoh Yi Shuen)