
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 \( a_{i} = \min \left\{ k + \frac{i}{k} \mid k \in \mathbf{N}^{*} \right\} \), find the value of \( S_{n}^{2} = \left[ a_{1} \right] + \left[ a_{2} \right] + \cdots + \left[ a_{n^{2}} \right] \), where \( n \geq 2 \) and \( [x] \) denotes the greatest integer less than or equal to \( x \).

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