$[40$ points$]$ Consider the search problem below with start state $\backslash ($ S $\backslash )$ and goal state $\backslash ($ G $\backslash ).$ The transition costs are next to the edges, and the heuristic values are next to the states. Show how the search proceeds step$-$by$-$step using the following algorithms. For full credit, you need to show how nodes are added and removed from the fringe, and which path to the goal is returned by each algorithm.

$($a$)$ Greedy Search

$($b$)$ A$*$ Search