Problem $3$: Dynamic Programming Minimum Cost Path through a Grid $(15$ points$)$

Given the grid shown below, determine the optimal path when starting at the goal $($z $=$ y$)$ and working

backward to the beginning point $($z $=$ a$),$ taking the lowest cost path at each decision point. What is the

final cost taking that approach? Now look for a lower cost by trying different paths. Is there a lower cost? Complete the table below which specifies the control action to take at each step, k$,$ when taking the

lowest cost route. Note that u $=1$ means going up and u $=-1$ means going down.