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 $n^{}_{}$.



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