
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
If all terms of the sequence $\left\{a_{n}\right\}$ are positive, and $\sqrt{a_{1}} + \sqrt{a_{2}} + \ldots + \sqrt{a_{n}} = n^2 + 3n\ (n \in \mathbb{N}^*)$, then $\lim\limits_{n \to \infty} \frac{1}{n^2} \left(\frac{a_{1}}{2} + \frac{a_{2}}{3} + \ldots + \frac{a_{n}}{n+1}\right) =$ \_\_\_\_\_\_\_\_\_\_\_\_\_.

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