Three positive real numbers $x, y, z$ satisfy $$\begin{aligned} x^{2} + y^{2} &= 3^{2} \\ y^{2} + yz + z^{2} &= 4^{2} \\ x^{2} + \sqrt{3}\,xz + z^{2} &= 5^{2} \end{aligned}$$ Find the value of $2xy + xz + \sqrt{3}\,yz$.