Use the two-phase simplex method to maximise $2 x_{1}+x_{2}+x_{3}$ subject to the constraints

$$x_{1}+x_{2} \geqslant 1, \quad x_{1}+x_{2}+2 x_{3} \leqslant 4, \quad x_{i} \geqslant 0 \text { for } i=1,2,3$$

Derive the dual of this linear programming problem and find the optimal solution of the dual.