A collection of 8 cubes consists of one cube with edge-length \(k\) for each integer \(k, 1 \le k \le 8.\) A tower is to be built using all 8 cubes according to the rules:
Any cube may be the bottom cube in the tower.
The cube immediately on top of a cube with edge-length \(k\) must have edge-length at most \(k+2.\)
Let \(T\) be the number of different towers than can be constructed. What is the remainder when \(T\) is divided by 1000?
(第二十四届AIME1 2006 第11题)