
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 positive integer with  $n \geq 2$ , and  $0<a_{1}, a_{2},...,a_{n}< 1$ . Find the maximum value of the sum
 $\sum_{i=1}^{n}(a_{i}(1-a_{i+1}))^{\frac{1}{6}}$ 
where  $a_{n+1}=a_{1}$ 

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