
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 \( n \) be a given positive integer, \( n \geq 2 \), and \( a_{1}, a_{2}, \cdots, a_{n} \in (0,1) \). Find the maximum value of \( \sum_{i=1}^{n} \sqrt[6]{a_{i}\left(1-a_{i+1}\right)} \), where \( a_{n+1} = a_{1} \).

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