Let $x > y > z > 0$ be real numbers.  Find the minimum value of
\[x + \frac{108}{(x - y)^3 (y - z)^2 z}.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $x > y > z > 0$ be real numbers.  Find the minimum value of
\[x + \frac{108}{(x - y)^3 (y - z)^2 z}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
7
Let $x > y > z > 0$ be real numbers.  Find the minimum value of
\[x + \frac{108}{(x - y)^3 (y - z)^2 z}.\]
train