Let $S(r)$ denote the sum of the infinite geometric series $17 + 17r + 17r^2 + 17r^3 + \ldots$ for $-1 < r < 1$. If $S(a) \times S(-a) = 2023$, find $S(a) + S(-a)$.

Let's think step by step and output the final answer within \boxed{}.
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