Three positive reals $x$, $y$, and $z$ satisfy the following equations:

1. $x^2 + y^2 = 9$  
2. $y^2 + yz + z^2 = 16$  
3. $x^2 + \sqrt{3}xz + z^2 = 25$

Find the value of $2xy + xz + \sqrt{3}yz$.

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