Let $\vartriangle ABC$ be an equilateral triangle with side length $12$. $D, E, F$ are points such that $ADEF$ is a square. The area of the possible geometric location of the centre of the square $ ADEF$ such the both segments $ED$ and $EF$ intersect the segment $BC$ (the possible geometric location of the centre is not a line or a curve) is $a\pi + b - c\sqrt{d}$, where $a$, $b$, $c$ and $d$ are positive integers and $d$ is square-free. Enter $a+b+c+d$.