How many integers \( n \) satisfy \( 1 \leq n \leq 100 \) such that \( \lfloor \sqrt{n} \rfloor^2 \leq n \leq \lfloor \sqrt{n} \rfloor^2 + 2\lfloor \sqrt{n} \rfloor \)? Let's think step by step and output the final answer within \boxed{}.
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