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Given that, $x$ and $y$ are non-negative integers. That means $x \geq 0, y \geq 0.$

And,

- $ x + 9 = z \Rightarrow x = z – 9 \; \longrightarrow (1) $
- $ y + 1 = z \Rightarrow y = z – 1 \; \longrightarrow (2) $
- $ x + y < z + 5 \; \longrightarrow (3) $

Put the value of $x,$ and $y$ in the equation $(3),$ we get.

$ x + y < z + 5 $

$ \Rightarrow (z – 9) + (z – 1) < z + 5 $

$ \Rightarrow z – 10 < 5 $

$ \Rightarrow \boxed{z < 15} $

Maximum value of $z$ can be $14.$

So,

- $ x_{\textsf{max}} = 14 – 9 = 5 $
- $ y_{\textsf{max}} = 14 – 1 = 13 $

Thus, the value of $2x+y = 2(5) + 13 = 23.$

$\therefore$ The maximum possible value of $2x+y$ is $23.$

Correct Answer$: 23 $