Question 37
$O$ is the center of square $ABCD$, and $M$ and $N$ are the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Points $A'$, $B'$, $C'$, $D'$ are chosen on $\overline{AO}$, $\overline{BO}$, $\overline{CO}$, $\overline{DO}$, respectively, so that $A'B'MC'D'N$ is an equiangular hexagon. The ratio $\frac{[A'B'MC'D'N]}{[ABCD]}$ can be written as $\frac{a+b\sqrt{c}}{d}$, where $a, b, c, d$ are integers, $d$ is positive, $c$ is square-free, and $\operatorname{gcd}(a, b, d)=1$. Find $1000a+100b+10c+d$.