Let $x$ and $y$ be real numbers.  Find the set of possible values of
\[\frac{(x + y)(1 - xy)}{(1 + x^2)(1 + y^2)}.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $x$ and $y$ be real numbers.  Find the set of possible values of
\[\frac{(x + y)(1 - xy)}{(1 + x^2)(1 + y^2)}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\left[ -\frac{1}{2}, \frac{1}{2} \right]
Let $x$ and $y$ be real numbers.  Find the set of possible values of
\[\frac{(x + y)(1 - xy)}{(1 + x^2)(1 + y^2)}.\]
train