Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 32.$  Find the minimum value of
\[x^2 + 4xy + 4y^2 + 2z^2.\]
Question:
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 32.$  Find the minimum value of
\[x^2 + 4xy + 4y^2 + 2z^2.\]
Answer:
Let's think step by step.

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Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 32.$  Find the minimum value of
\[x^2 + 4xy + 4y^2 + 2z^2.\]
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