State and prove the Lagrangian sufficiency theorem.

Solve the problem

$$\begin{array}{ll} \text { maximize } & x_{1}+3 \ln \left(1+x_{2}\right) \\ \text { subject to } \quad & 2 x_{1}+3 x_{2} \leqslant c_{1} \\ & \ln \left(1+x_{1}\right) \geqslant c_{2}, \quad x_{1} \geqslant 0, x_{2} \geqslant 0 \end{array}$$

where $c_{1}$ and $c_{2}$ are non-negative constants satisfying $c_{1}+2 \geqslant 2 e^{c_{2}}$.