Let $x,$ $y,$ and $z$ be positive real numbers.  Find the minimum value of
\[\frac{x^3 + 2y^3 + 4z^3}{xyz}.\]
<|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
\[\frac{x^3 + 2y^3 + 4z^3}{xyz}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
6
Let $x,$ $y,$ and $z$ be positive real numbers.  Find the minimum value of
\[\frac{x^3 + 2y^3 + 4z^3}{xyz}.\]
train