Let $w,$ $x,$ $y,$ and $z,$ be positive real numbers.  Find the maximum value of
\[\frac{wx + xy + yz}{w^2 + x^2 + y^2 + z^2}.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $w,$ $x,$ $y,$ and $z,$ be positive real numbers.  Find the maximum value of
\[\frac{wx + xy + yz}{w^2 + x^2 + y^2 + z^2}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{1 + \sqrt{5}}{4}
Let $w,$ $x,$ $y,$ and $z,$ be positive real numbers.  Find the maximum value of
\[\frac{wx + xy + yz}{w^2 + x^2 + y^2 + z^2}.\]
train