Let $a$, $b$, $c$, and $d$ be positive real numbers such that:
\[
\begin{align*}
a^2 + b^2 &= c^2 + d^2 = 2008, \\
ac &= bd = 1000.
\end{align*}
\]
If $S = a + b + c + d$, compute the value of $\lfloor S \rfloor$.

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