 Solution

The duplicated feasible set is the diamond $|{\beta }_1|+|{\beta }_2|\le t$. A loss contour first touches an entire line segment of the diamond whenever the optimal total has the sign of a sloping face; all points on that segment give the same fit. As $t$ passes the unconstrained optimum, the solution set becomes the intersection of the interior diamond with the line ${\beta }_1+{\beta }_2=z/s$.