
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
Let \( n \in \mathbf{Z}_{+} \). Does there exist a positive real number \(\varepsilon=\varepsilon(n)\) such that for any \( x_{1}, x_{2}, \cdots, x_{n} > 0 \), the following inequality holds?
$$
\begin{array}{l}
\left(\prod_{i=1}^{n} x_{i}\right)^{\frac{1}{n}} \\
\leqslant(1-\varepsilon) \frac{1}{n} \sum_{i=1}^{n} x_{i}+\varepsilon n \left(\sum_{i=1}^{n} \frac{1}{x_{i}}\right)^{-1}
\end{array}
$$

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