Explain how Benders decomposition work in general to solve mixed integer programming problems
(MIP). Solve the MIP below with this method. Show all your steps.
Max $z = 2x_1 + 4x_2 + 5x_3 + 3y_1 + 3y_2 + 5y_3$
s.t. $x_1 + 3x_2 + 4x_3 + 2y_1 + y_2 + 2y_3 \le 67$
$2x_1 + 2x_2 + 2x_3 + y_1 + 2y_2 + y_3 \le 52$
$x_1 + x_2 + x_3 + 2y_1 + y_2 + 3y_3 \le 53$
$x_3 + y_1 + 2y_2 + 3y_3 \le 40$
$x_1, x_2, x_3 \ge 0$
$y_1, y_2, y_3 \in \mathbb{Z}^0_+$
The set $\mathbb{Z}^0_+$ denotes the set of non-negative integers.