
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
Let \( n \) be a positive integer, and denote by \( f(n) \) the smallest positive integer \( k \) for which \( \left\lfloor \frac{n^2}{k} \right\rfloor = \left\lfloor \frac{n^2}{k+1} \right\rfloor \). Determine the value of \( \left\lfloor \frac{n^2}{f(n)} \right\rfloor + f(n) \).

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