Find the real number $k$ such that $a$, $b$, $c$, and $d$ are real numbers that satisfy the system of equations:
\[
\begin{align*}
    abcd &= 2007, \\
    a &= \sqrt{55 + \sqrt{k+a}}, \\
    b &= \sqrt{55 - \sqrt{k+b}}, \\
    c &= \sqrt{55 + \sqrt{k-c}}, \\
    d &= \sqrt{55 - \sqrt{k-d}}.
\end{align*}
\]

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