A Minimization that is not so Clear

Algebra Level 5

For a positive integer \(n\), define \(S_n\) to be the minimum value of the sum \[ \sum_{k = 1}^n \sqrt{(2k-1)^2 + (a_k)^2}, \] where \(a_1, a_2,\ldots, a_n\) are positive real numbers whose sum is \(17\). There is a unique positive integer \(n\) for which \(S_n\) is also an integer. Find this \(n\).

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