HMMT 2024 G7

Let $A B C$ be an acute triangle. Let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $\overline{B C}$, $\overline{C A}$, and $\overline{A B}$, respectively, and let $Q$ be the foot of altitude from $A$ to line $E F$. Given that $A Q=20$, $B C=15$, and $A D=24$, compute the perimeter of triangle $D E F$.