Question 89
Let $\triangle ABC$ be an equilateral triangle with side length $16$. Points $D, E, F$ are on $CA, AB$, and $BC$, respectively, such that $DE \perp AE$, $DF \perp CF$, and $BD = 14$. The perimeter of $\triangle BEF$ can be written in the form $a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6}$, where $a, b, c$, and $d$ are integers. Find $a + b + c + d$.