$.$ Implement a stack data structure using Java that supports the following operations:

$1.$ push$($item$)$: Adds an item to the top of the stack

$2.$ pop$()$: Removes and returns the item at the top of the stack

$3.$ peek$()$: Returns the item at the top of the stack without removing it$.$

$4.$ is$\_$empty$()$: Returns True if the stack is empty, False otherwise

Now, extend your stack implementation to include a new operation:

$5.$ get$\_$min$()$: Returns the minimum item in the stack without removing it$.$

Explain how will you ensure that this operation has a time complexity of O$(1)$

B$.$ Provide the code for your stack implementation and demonstrate how you would use it to push and pop items, retrieve the top item, check if the stack is empty, and efficiently retrieve the minimum item.

C$.$ Test your stack implementation with various scenarios like queue and priority queue to ensure its correctness and efficiency, especially for the get$\_$min$()$ operation.