Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a^2 + b^2 + c^2 = 1.$  Find the maximum value of
\[2ab \sqrt{2} + 2bc.\]
Question:
Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a^2 + b^2 + c^2 = 1.$  Find the maximum value of
\[2ab \sqrt{2} + 2bc.\]
Answer:
Let's think step by step.

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Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a^2 + b^2 + c^2 = 1.$  Find the maximum value of
\[2ab \sqrt{2} + 2bc.\]
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