Let $S(k)$ be the sum of three or more distinct positive integers having the following properties:

• No value exceeds $k$.
• The values form a geometric progression.
• The sum is maximal.

$S(4)=4+2+1=7$
$S(10)=9+6+4=19$
$S(12)=12+6+3=21$
$S(1000)=1000+900+810+729=3439$

Let $T(n)=\sum _{k=4}^n(-1{)}^{k}S(k)$.
$T(1000)=2268$

Find $T({10}^{17})$.