
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 positive integer \( n \), there are \( 3n \) numbers that satisfy the following conditions:
$$
\begin{array}{c}
0 \leqslant a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{3n}, \\
\text { and }\left(\sum_{i=1}^{3n} a_{i}\right)^{3} \geqslant k_{n}\left(\sum_{i=1}^{n} a_{i} a_{n+i} a_{2n+i}\right) \text { always holds. }
\end{array}
$$

Find the optimal value of \( k_{n} \) in terms of \( n \).

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