State and prove the Lagrangian sufficiency theorem.

Solve, using the Lagrangian method, the optimization problem

$$\begin{array}{ll} \operatorname{maximise} & x+y+2 a \sqrt{1+z} \\ \text { subject to } & x+\frac{1}{2} y^{2}+z=b \\ & x, z \geqslant 0 \end{array}$$

where the constants $a$ and $b$ satisfy $a \geqslant 1$ and $b \geqslant 1 / 2$.

[You need not prove that your solution is unique.]