Let $S(n)=\sum a+b+c$ over all triples $(a,b,c)$ such that:

• $a$, $b$ and $c$ are prime numbers.
• $a<b<c<n$.
• $a+1$, $b+1$, and $c+1$ form a geometric sequence.

For example, $S(100)=1035$ with the following triples:

$(2,5,11)$, $(2,11,47)$, $(5,11,23)$, $(5,17,53)$, $(7,11,17)$, $(7,23,71)$, $(11,23,47)$, $(17,23,31)$, $(17,41,97)$, $(31,47,71)$, $(71,83,97)$

Find $S({10}^8)$.