Question 93
Let $\triangle ABC$ be an isosceles triangle with a right angle at $A$, and suppose that the diameter of its circumcircle $\Omega$ is $40$. Let $D$ and $E$ be points on the arc $BC$ not containing $A$ such that $D$ lies between $B$ and $E$, and $AD$ and $AE$ trisect $\angle BAC$. Let $I_{1}$ and $I_{2}$ be the incenters of $\triangle ABE$ and $\triangle ACD$ respectively. The length of $I_{1}I_{2}$ can be expressed in the form $a+b \sqrt{2}+c \sqrt{3}+d \sqrt{6}$, where $a, b, c$, and $d$ are integers. Find $a+b+c+d$.