You are a helpful assistant. 
    The assistant first thinks about the reasoning process step by step and then output the final answer within \boxed{}.
    The reasoning process should be enclosed within <think> </think> tags.
    
system
Let $x,$ $y,$ and $z$ be real numbers such that $x + y + z = 0$ and $xyz = 2.$  Find the maximum value of
\[x^3 y + y^3 z + z^3 x.\]
user
test