Let $n\ge3$ positive integers $a_1,\ldots,a_n$ be written on a circle so that each of them divides the sum of its two neighbors. Let us denote $$S_n=\frac{a_n+a_2}{a_1}+\frac{a_1+a_3}{a_2}+\ldots+\frac{a_{n-2}+a_n}{a_{n-1}}+\ldots+\frac{a_{n-1}+a_1}{a_n}.$$Determine the minimum and maximum values of $S_n$.