You are tasked with designing an algorithm to support a special stack that, in addition to standard push and pop operations, has a function min$()$ which returns the minimum element in the stack. All operations $($push$,$ pop, and min$)$ should be performed in constant time, O$(1).$ To achieve this, you propose using an auxiliary stack that keeps track of the minimum elements. The primary stack $($mainStack$)$ is used for standard stack operations, while the auxiliary stack $($minStack$)$ is used to keep track of minimum values. Here is how the algorithm works:

Push$($x$)$: Push x onto mainStack. If minStack is empty or x is less than or equal to the top element of minStack, push x onto minStack as well.

Pop$()$: Pop the top element from mainStack. If the popped element is the same as the top element of minStack, pop the top element from minStack too.

Min$()$: Return the top element of minStack, which represents the current minimum in mainStack.

Which of the following statements is correct about the proposed algorithm?

Question $2$ options:

The algorithm does not correctly return the minimum element if the min$()$ operation is called after a series of pop operations.

The algorithm's min$()$ operation has a time complexity of O$($n$),$ where nn is the number of elements in the stack.

The algorithm successfully implements the special stack with all operations in O$(1)$ time, but it uses extra space proportional to the size of the input.

The algorithm fails to maintain the O$(1)$ time complexity for the min$()$ operation if there are duplicate minimum elements in the stack.