For 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 \(n^{}_{}\).

(第九届AIME1991 第15题)