
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 \( n \ge 2 \), define \( A(n) \) as the area of the region in the coordinate plane bound by \( 1 \le x \le n \) and \( 0 \le y \le x\lfloor \sqrt{x} \rfloor \). Calculate how many integers \( n \) within the range \( 2 \le n \le 2000 \) make \( A(n) \) an integer.

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