
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

Given a real number \( a > 25 \), for all positive real number arrays \(\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)\) satisfying the condition \(\sum_{i=1}^{5} x_{i} \cdot \sum_{i=1}^{5} \frac{1}{x_{i}} = a\), find the extreme value of \(\frac{\max \left\{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right\}}{\min \left\{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right\}}\).

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