Determine all real values of $A$ for which there exist distinct complex numbers $x_1$, $x_2$ such that the following three equations hold:
\[
x_1(x_1+1) = A \\
x_2(x_2+1) = A \\
x_1^4 + 3x_1^3 + 5x_1 = x_2^4 + 3x_2^3 + 5x_2.
\]

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