
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
Suppose $n \geq 2$ is a positive integer and $a_1, a_2, \ldots, a_n$ are non-negative real numbers such that $\sum_{i=1}^{n} a_i = 1$. Find the maximum value of $\left( \sum_{i=1}^{n} i^2 a_i \right) \cdot \left( \sum_{i=1}^{n} \frac{a_i}{i} \right)^{2}$.

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