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518. Prime triples and geometric sequences

Let S(n) = Σ 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(10⁸).