
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 that $n$ and $k$ are positive integers with $n > k$, and real numbers $a_{1}, a_{2}, \cdots, a_{n} \in (k-1, k)$. Let positive real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfy the condition that for any $k$-element subset $I$ of $\{1,2,\cdots, n\}$, the inequality $\sum_{i \in I} x_{i} \leqslant \sum_{i \in I} a_{i}$ holds. Find the maximum value of $x_{1} x_{2} \cdots x_{n}$.

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