HMMT 2024 U30

Let $A B C$ be an equilateral triangle with side length 1 . Points $D, E, F$ lie inside triangle $A B C$ such that $A, E, F$ are collinear, $B, F, D$ are collinear, $C, D, E$ are collinear, and triangle $D E F$ is equilateral. Suppose that there exists a unique equilateral triangle $X Y Z$ with $X$ on side $\overline{B C}, Y$ on side $\overline{A B}$, and $Z$ on side $\overline{A C}$ such that $D$ lies on side $\overline{X Z}, E$ lies on side $\overline{Y Z}$, and $F$ lies on side $\overline{X Y}$. Compute $A Z$.