$12.$ Consider the following graph, where nodes represent cities, and the edges represent the roads between them.

i$.$ Provided the cost of travel, obtain the optimal path from the initial node to the Goal node using the Best First Search approach.

ii$.$ Provided the heuristic values, obtain the optimal path from the initial node to the Goal node using the A$-$star search approach.

iii. Traverse the graph from the initial node to the Goal node using: $($a$)$ Breadth First Search and $($b$)$ Depth First Search technique. Show the steps.