Let $a,$ $b,$ and $c$ be real numbers such that $a \ge b \ge c \ge 0$ and $a + b = 16.$  Find the maximum value of
\[\sqrt{c(b - c)} + \sqrt{c(a - c)}.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $a,$ $b,$ and $c$ be real numbers such that $a \ge b \ge c \ge 0$ and $a + b = 16.$  Find the maximum value of
\[\sqrt{c(b - c)} + \sqrt{c(a - c)}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
8
Let $a,$ $b,$ and $c$ be real numbers such that $a \ge b \ge c \ge 0$ and $a + b = 16.$  Find the maximum value of
\[\sqrt{c(b - c)} + \sqrt{c(a - c)}.\]
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