CEMC Fryer, 2005

Points $B$, $C$, and $D$ lie on a line segment $AE$, as shown. The line segment $AE$ has 4 basic sub-segments $AB$, $BC$, $CD$, and $DE$, and 10 sub-segments: $AB$, $AC$, $AD$, $AE$, $BC$, $BD$, $BE$, $CD$, $CE$, and $DE$. The super-sum of $AE$ is the sum of the lengths of all of its sub-segments. If $AB = 3$, $BC = 6$, $CD = 9$, and $DE = 7$, determine the lengths of the 10 sub-segments, and calculate the super-sum of $AE$.