
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 the sequence $\{a_n\}$ and the sequence $\left\{\frac{a_n^2}{n}\right\}$ are both arithmetic progressions $(n\in\mathbb{N}^*)$, and $a_1=2$, find the sum $a_1 + \left(\frac{a_2}{2}\right)^2 + \left(\frac{a_3}{3}\right)^3 + \ldots + \left(\frac{a_n}{n}\right)^n$.

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