Find the number of positive integers $n$ between $1$ and $1000$, inclusive, satisfying
\[ \lfloor \sqrt{n - 1} \rfloor + 1 = \left\lfloor \sqrt{n + \sqrt{n} + 1}\right\rfloor\]
where $\lfloor n \rfloor$ denotes the greatest integer not exceeding $n$.

Let's think step by step and output the final answer within \boxed{}.
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