
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 the sequence $\{a_n\}$ consists of positive terms, and $\sqrt {a_{1}} + \sqrt {a_{2}} + \ldots + \sqrt {a_{n}} = n^2 + 3n$ for $n$ belonging to the natural numbers, find the limit $\lim_{n \to \infty} \left( \frac {1}{n^{2}} \left( \frac {a_{1}}{2} + \frac {a_{2}}{3} + \ldots + \frac {a_{n}}{n+1} \right) \right)$.

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