$7.(10$ marks$)$ Consider the following story: Once upon a time a farmer went to market and purchased a fox, a goose, and a bag of beans. On his way home, the farmer came to the bank of a river and hired a boat. But in crossing the river by boat, the farmer could carry only himself and a single one of his purchases $-$ the fox, the goose, or the hag of the beans.

If left alone, the fox would eat the goose, and the goose would eat the beans.

The farmer's challenge was to carry himself and his purchases to the far bank of the river, leaving each purchase intact. How did he do it$?$

Use state$-$space$-$search method to solve this problem. Draw EXPLICITLY the state space and give the solution based on both DFS and BFS$.$