Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 5.$  Find the maximum value of
\[\sqrt{2x + 1} + \sqrt{2y + 1} + \sqrt{2z + 1}.\]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Let's think step by step and output the final answer within \boxed{}.
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