65. Let $a_{1}, a_{2}, \cdots, a_{n}$ be positive real numbers, and $n>2$ be a given positive integer. Find the largest positive number $K$ and the smallest positive number $G$ such that the following inequality holds: $K<\frac{a_{1}}{a_{1}+a_{2}}+\frac{a_{2}}{a_{2}+a_{3}}+\cdots+\frac{a_{n}}{a_{n}+a_{1}}<G$. (1991 Japan Mathematical Olympiad)
65. Let $a_{n+1}=a_{1}, S_{n}=\frac{a_{1}}{a_{1}+a_{2}}+\frac{a_{2}}{a_{2}+a_{3}}+\cdots+\frac{a_{n}}{a_{n}+a_{1}}$, then
$$S_{n}>\frac{a_{1}}{a_{1}+a_{2}+\cdots+a_{n}}+\frac{a_{2}}{a_{1}+a_{2}+\cdots+a_{n}}+\cdots+\frac{a_{n}}{a_{1}+a_{2}+\cdots+a_{n}}=1$$

Also,
$$\begin{aligned}
S_{n}+1= & S_{n}+\frac{a_{1}}{a_{1}+a_{2}+\cdots+a_{n}}+\frac{a_{2}}{a_{1}+a_{2}+\cdots+a_{n}}+\cdots+\frac{a_{n}}{a_{1}+a_{2}+\cdots+a_{n}}0$, then we have $S_{n}=\frac{n-1}{1+t}+\frac{t^{n-1}}{1+t^{n-1}}$.
Since $\lim _{t \rightarrow 0} S_{n}(t)=n-1, \lim _{t \rightarrow+\infty} S_{n}(t)=1$, the bounds of the inequality are optimal.