Question 116
Let the $\triangle ABC$ with $AB = 1\ \text{cm}$, $BC = 2\ \text{cm}$, and $AC = \sqrt{3}\ \text{cm}$ be given. The points $D, E$, and $F$ lie on the sides $AB, AC$, and $BC$, respectively, satisfying $AE = BD$ and $BF = AD$. The angle bisector of $\angle BAC$ intersects for the second time the circle through $A, D$, and $E$ at $M$, while the angle bisector of $\angle ABC$ intersects for the second time the circle through $B, D$, and $F$ at $N$. Find the length of the line segment $MN$.