Solve the following Optimization problem using the simplex algorithm:

$$\begin{array}{rr} \operatorname{maximise} & x_{1}+x_{2} \\ \text { subject to } & \left|x_{1}-2 x_{2}\right| \leqslant 2 \\ & 4 x_{1}+x_{2} \leqslant 4, \quad x_{1}, x_{2} \geqslant 0 \end{array}$$

Suppose the constraints above are now replaced by $\left|x_{1}-2 x_{2}\right| \leqslant 2+\epsilon_{1}$ and $4 x_{1}+x_{2} \leqslant 4+\epsilon_{2}$. Give an expression for the maximum objective value that is valid for all sufficiently small non-zero $\epsilon_{1}$ and $\epsilon_{2}$.