please i do not need any AI answer $.$ please solve by subject experts. Finding shortest path

Consider the problem of finding the shortest path between two points on a plane that has convex polygonal obstacles, as shown below. $S$ and $G$ are the start and goal states. This is an idealization of the problem that a robot has to solve to navigate in a crowded environment.

Suppose the state space consists of all positions $(x,y)$ in the plane. How many states are there? How many paths are there to the goal?

Answer:

$2.$ Explain briefly why the shortest path from one polygon vertex to any other in the scene must consist of straight$-$line segments joining some of the vertices of the polygons. Define a good state space now. How large is this state space?

$3.$ Define the necessary functions to implement the search problem, including an ACTIONS function that takes a vertex as input and returns a set of vectors, each of which maps the current vertex to one of the vertices that can be reached in a straight line. $($Do not forget the neighbors on the same polygon.$)$ Use the straight$-$line distance for the heuristic function.