$\square$ Example 8 Given an integer $n$ greater than 3, let real numbers $x_{1}, x_{2}, \cdots, x_{n}, x_{n+1}, x_{n+2}$ satisfy $0<x_{1}<x_{2}<\cdots<x_{n}<x_{n+1}<x_{n+2}$. Try to find the minimum value of
$$\frac{\left(\sum_{i=1}^{n} \frac{x_{i+1}}{x_{i}}\right)\left(\sum_{j=1}^{n} \frac{x_{j+2}}{x_{j+1}}\right)}{\left(\sum_{k=1}^{n} \frac{x_{k+1} x_{k+2}}{x_{k+1}^{2}+x_{k+1} x_{k+2}}\right)\left(\sum_{i=1}^{n} \frac{x_{i+1}^{2}+x_{t} x_{k+2}}{x_{l} x_{k+2}}\right)}$$

and find all real number sequences $x_{1}, x_{2}, \cdots, x_{n}, x_{n+1}, x_{n+2}$ that satisfy the conditions and make the expression achieve its minimum value. (2001 China National Training Team Selection Exam Problem)
Let $t_{i}=\frac{x_{i+1}}{x_{i}}(>1), 1 \leqslant i \leqslant n$, the expression in the problem can be written as
$$\frac{\left(\sum_{i=1}^{n} t_{i}\right)\left(\sum_{i=1}^{n} t_{i+1}\right)}{\left(\sum_{i=1}^{n} \frac{t_{i} t_{i+1}}{t_{i}+t_{i+1}}\right)\left[\sum_{i=1}^{n}\left(t_{i}+t_{i+1}\right)\right]}$$

By the Cauchy-Schwarz inequality, we have
$$\begin{aligned}
& \left(\sum_{i=1}^{n} \frac{t_{i} t_{i+1}}{t_{i}+t_{i+1}}\right)\left[\sum_{i=1}^{n}\left(t_{i}+t_{i+1}\right)\right] \\
= & \left(\sum_{i=1}^{n} t_{i}-\sum_{i=1}^{n} \frac{t_{i}^{2}}{t_{i}+t_{i+1}}\right)\left[\sum_{i=1}^{n}\left(t_{i}+t_{i+1}\right)\right] \\
= & \left(\sum_{i=1}^{n} t_{i}\right)\left[\sum_{i=1}^{n}\left(t_{i}+t_{i+1}\right)\right]-\left(\sum_{i=1}^{n} \frac{t_{i}^{2}}{t_{i}+t_{i+1}}\right)\left[\sum_{i=1}^{n}\left(t_{i}+t_{i+1}\right)\right] \\
\leqslant & \left(\sum_{i=1}^{n} t_{i}\right)\left[\sum_{i=1}^{n}\left(t_{i}+t_{i+1}\right)\right]-\left(\sum_{i=1}^{n} \frac{t_{i}}{\sqrt{t_{i}+t_{i+1}}} \cdot \sqrt{t_{i}+t_{i+1}}\right)^{2} \\
= & \left(\sum_{i=1}^{n} t_{i}\right)^{2}+\left(\sum_{i=1}^{n} t_{i}\right)\left(\sum_{i=1}^{n} t_{i+1}\right)-\left(\sum_{i=1}^{n} t_{i}\right)^{2} \\
= & \left(\sum_{i=1}^{n} t_{i}\right)\left(\sum_{i=1}^{n} t_{i+1}\right)
\end{aligned}$$

Therefore, for any real number sequence $x_{1}, x_{2}, \cdots, x_{n}, x_{n+1}, x_{n+2}$ that satisfies the conditions, the expression in the problem is no less than 1.