Let $x,$ $y,$ $z$ be positive real numbers such that $x + y + z = 9.$  Find the maximum value of
\[\sqrt{16 - x^2} + \sqrt{25 - y^2} + \sqrt{36 - z^2}.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $x,$ $y,$ $z$ be positive real numbers such that $x + y + z = 9.$  Find the maximum value of
\[\sqrt{16 - x^2} + \sqrt{25 - y^2} + \sqrt{36 - z^2}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
12
Let $x,$ $y,$ $z$ be positive real numbers such that $x + y + z = 9.$  Find the maximum value of
\[\sqrt{16 - x^2} + \sqrt{25 - y^2} + \sqrt{36 - z^2}.\]
train