Use the two phase method to find all optimal solutions to the problem

$$\begin{aligned} \operatorname{maximize} 2 x_{1}+3 x_{2}+x_{3} \\ \text { subject to } x_{1}+x_{2}+x_{3} & \leqslant 40 \\ 2 x_{1}+x_{2}-x_{3} & \geqslant 10 \\ -x_{2}+x_{3} & \geqslant 10 \\ x_{1}, x_{2}, x_{3} & \geqslant 0 \end{aligned}$$

Suppose that the values $(40,10,10)$ are perturbed to $(40,10,10)+\left(\epsilon_{1}, \epsilon_{2}, \epsilon_{3}\right)$. Find an expression for the change in the optimal value, which is valid for all sufficiently small values of $\epsilon_{1}, \epsilon_{2}, \epsilon_{3}$.

Suppose that $\left(\epsilon_{1}, \epsilon_{2}, \epsilon_{3}\right)=(\theta,-2 \theta, 0)$. For what values of $\theta$ is your expression valid?