The polynomial $f(x) = x^{4} + ax^{3} + bx^{2} + cx + d$ has real coefficients, and $f(2i) = f(2 + i) = 0$. What is $a + b + c + d$?

$\textbf{(A)}\ 0 \ \textbf{(B)}\ 1 \ \textbf{(C)}\ 4 \ \textbf{(D)}\ 9 \ \textbf{(E)}\ 16$

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