Let $x$, $y$, $z$ be real numbers that satisfy the following equations:

\[
(x-y)^2 + (y-z)^2 + (z-x)^2 = 8, \\
x^3 + y^3 + z^3 = 1
\]

Find the minimum value of $x^4 + y^4 + z^4$.

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