
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 \( n \) is a specified positive integer not less than 2, find the maximum value of the expression 

\[
\frac{\left(\sum_{i=1}^{n} i a_{i}\right)\left(\sum_{i=1}^{n} \frac{a_{i}}{i}\right)}{\left(\sum_{i=1}^{n} a_{i}\right)^{2}}
\]

for all non-negative arrays \( a_{i} \) (for \( i=1, 2, \cdots, n \)).

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