Let $x,$ $y,$ and $z$ be real numbers such that $x + y + z = 0$ and $xyz = 2.$  Find the maximum value of
\[x^3 y + y^3 z + z^3 x.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $x,$ $y,$ and $z$ be real numbers such that $x + y + z = 0$ and $xyz = 2.$  Find the maximum value of
\[x^3 y + y^3 z + z^3 x.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
-9
Let $x,$ $y,$ and $z$ be real numbers such that $x + y + z = 0$ and $xyz = 2.$  Find the maximum value of
\[x^3 y + y^3 z + z^3 x.\]
train