$(10$ points$)$ Consider the following problem, and answer the questions.

$mi{n}_{x_1,x_2}02x_1+3x_2+2$max$\{(3-x_1-2x_2),(\frac{4}{3}-\frac{2x_1}{3}+\frac{x_2}{3}),0\}.$

$($a$)(4$ points$)$ Reformulate $(1)$ into a LP by introducing one additional variable $y.$

$($b$)(2$ points$)$ Show that your LP reformulation is indexd equivalent to $(1).$

$($c$)(4$ points$)$ Solve the linearized LP using Benders decomposition by treating $(\{$:$x_1,x_2)$ as the

first stage variables and $y$ as the second stape variable.