
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 all terms in the sequence $\{a\_n\}$ are positive numbers, and $\sqrt{a\_1} + \sqrt{a\_2} + \cdots + \sqrt{a\_n} = n^2 + 3n (n \in \mathbb{N}^*)$, find the limit $\lim\_{n \to \infty} \frac{1}{n^2} \left( \frac{a\_1}{2} + \frac{a\_2}{3} + \cdots + \frac{a\_n}{n+1} \right) = $.

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