Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x^2 + 2y^2 + 5z^2 = 22.$  Find the maximum value of
\[xy + xz + yz.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x^2 + 2y^2 + 5z^2 = 22.$  Find the maximum value of
\[xy + xz + yz.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
11
Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x^2 + 2y^2 + 5z^2 = 22.$  Find the maximum value of
\[xy + xz + yz.\]
train