Is this even a pattern?

Let \(S_{1}\) represent \(1^{2+3}\) and \(S_{2}\) represent \(2^{3+4}\) and this continues to the infinity. Given that \(a_{1}\) is the digit sum of \(S_{999}\) and \(a_{2}\) is the digit sum of \(a_{1}\) and so on and so forth, find the value of \(a_{99999999}\)

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This question is based on a popular number theory question.

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