Suppose $x, y \in \mathbb{Z}$ satisfy the equation:
\[
y^4 + 4y^3 + 28y + 8x^3 + 6y^2 + 32x + 1 = (x^2 - y^2)(x^2 + y^2 + 24).
\]
Find the sum of all possible values of $|xy|$.

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