
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
The first term of a sequence is \( a_{1} = 1 \), and each subsequent term is defined by

\[ a_{n+1} = 1 + \frac{n}{a_{n}}, \quad n = 1, 2, 3, \ldots \]

Does the following limit exist? If it exists, determine it.

\[ \lim_{n \rightarrow \infty} \left(a_{n} - \sqrt{n}\right) \]

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