
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]


system
For a given natural number  $n > 3$ , the real numbers  $x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2}$  satisfy the conditions  $0
< x_1 < x_2 < \cdots < x_n < x_{n + 1} < x_{n + 2}$ .  Find the minimum possible value of
\[\frac{(\sum _{i=1}^n \frac{x_{i + 1}}{x_i})(\sum _{j=1}^n \frac{x_{j + 2}}{x_{j +
1}})}{(\sum _{k=1}^n \frac{x_{k + 1} x_{k + 2}}{x_{k + 1}^2 + x_k
x_{k + 2}})(\sum _{l=1}^n \frac{x_{l + 1}^2 + x_l x_{l + 2}}{x_l
x_{l + 1}})}\] and find all  $(n + 2)$ -tuplets of real numbers  $(x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2})$  which gives this value.

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