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

user
\sqrt{2}
Let $x,$ $y,$ and $z$ be positive real numbers.  Find the minimum value of
\[\frac{\sqrt{x^2 + y^2} + \sqrt{x^2 + z^2} + \sqrt{y^2 + z^2}}{x + y + z}.\]
train