
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  $a_i=min\{ k+\dfrac{i}{k}|k \in N^*\}$ , determine the value of  $S_{n^2}=[a_1]+[a_2]+\cdots +[a_{n^2}]$ , where  $n\ge 2$  . ( $[x]$  denotes the greatest integer not exceeding x)

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