$.$ Suppose we are given a matrix A in Rm$$n and a vector b in Rm such that b $0.$ We are interested

in an algorithm that, given this A and b$,$ is guaranteed to produce one of following two outputs:

$($i$)$ If there exists x such that Ax $$ b$,$ then the algorithm produces one such x$.($ii$)$ If no such x

exists, then the algorithm produces an output to declare so$.$

Describe in detail how to design this algorithm based on the simplex method.