Suppose we have a sequence $a_1, a2_, ...$ of positive real numbers so that for each positive integer $n$, we have that $\sum_{k=1}^{n} a_ka_{\lfloor \sqrt{k} \rfloor} = n^2$. Determine the first value of $k$ so $a_k > 100$.

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