Example 3.15 (2007 Girls' Mathematical Olympiad) Let $n>3$ be an integer, and let $a_{1}, a_{2}, \cdots, a_{n}$ be non-negative real numbers satisfying $a_{1}+a_{2}+\cdots+a_{n}=2$. Find the minimum value of
$$\frac{a_{1}}{a_{2}^{2}+1}+\frac{a_{2}}{a_{3}^{2}+1}+\cdots+\frac{a_{n}}{a_{1}^{2}+1}.$$