
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:
\[ 
0 \leq a_{1} \leq a_{2} \leq \cdots \leq a_{3n}, 
\]
and
\[ 
\left(\sum_{i=1}^{3n} a_{i}\right)^{3} \geq k_{n}\left(\sum_{i=1}^{n} a_{i} a_{n+i} a_{2n+i}\right) \text{ always holds. }
\]

Find the best possible value of \( k_{n} \) (expressed in terms of \( n \)).

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