
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 $n \geq 2$, define $B(n)$ as the area of the region in the coordinate plane determined by the inequalities $2 \leq x \leq n$ and $0 \leq y \leq x \lceil \sqrt{x} \rceil$, where $\lceil \sqrt{x} \rceil$ is the smallest integer not less than $\sqrt{x}$. Find the number of integer values of $n$, between $2$ and $2000$, for which $B(n)$ is an integer.

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