
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


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Given an integer $n$ greater than 3, let real numbers $x_{1}, x_{2}, \cdots, x_{n}, x_{n+1}, x_{n+2}$ satisfy the conditions $0 < x_{1} < x_{2} < \cdots < x_{n} < x_{n+1} < x_{n+2}$. Find the minimum value of the expression

\[
\frac{\left(\sum_{i=1}^{n} \frac{x_{i+1}}{x_{i}}\right)\left(\sum_{j=1}^{n} \frac{x_{j+2}}{x_{j+1}}\right)}{\left(\sum_{k=1}^{n} \frac{x_{k+1} x_{k+2}}{x_{k+1}^{2}+x_{k} x_{k+2}}\right)\left(\sum_{i=1}^{n} \frac{x_{l+1}^{2}+x_{l} x_{l+2}}{x_{l} x_{l+1}}\right)},
\]

and determine the sequences $x_{1}, x_{2}, \cdots, x_{n}, x_{n+1}, x_{n+2}$ that achieve this minimum value.

Present the answer in LaTex format: \boxed{Your answer}
user