Example 38 (1982 West German Mathematical Olympiad) Non-negative real numbers $a_{1}, a_{2}, \cdots, a_{n}$ satisfy $\sum_{k=1}^{n} a_{k}=1$, find the minimum value of $\frac{a_{1}}{1+a_{2}+a_{3}+\cdots+a_{n}}+\frac{a_{2}}{1+a_{1}+a_{3}+\cdots+a_{n}}+\cdots+\frac{a_{n}}{1+a_{1}+a_{2}+\cdots+a_{n-1}}$.