If $x,$ $y,$ and $z$ are positive real numbers such that
\[\frac{x + 2y + 4z + 8}{4} = \frac{4}{\frac{1}{x} + \frac{1}{2y} + \frac{1}{4z} + \frac{1}{8}}.\]Find the ordered triple $(x,y,z).$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
If $x,$ $y,$ and $z$ are positive real numbers such that
\[\frac{x + 2y + 4z + 8}{4} = \frac{4}{\frac{1}{x} + \frac{1}{2y} + \frac{1}{4z} + \frac{1}{8}}.\]Find the ordered triple $(x,y,z).$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
(8,4,2)
If $x,$ $y,$ and $z$ are positive real numbers such that
\[\frac{x + 2y + 4z + 8}{4} = \frac{4}{\frac{1}{x} + \frac{1}{2y} + \frac{1}{4z} + \frac{1}{8}}.\]Find the ordered triple $(x,y,z).$
train