
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
For each integer  $n \ge 2$ , let  $A(n)$  be the area of the region in the coordinate plane defined by the inequalities  $1\le x \le n$  and  $0\le y \le x \left\lfloor \sqrt x \right\rfloor$ , where  $\left\lfloor \sqrt x \right\rfloor$  is the greatest integer not exceeding  $\sqrt x$ . Find the number of values of  $n$  with  $2\le n \le 1000$  for which  $A(n)$  is an integer.

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