Imagine there is a modified version of many-to-one school choice matching with Deferred Acceptance Algorithm(DAA): other things will be the same as original DAA, except that for an unassigned student, who was rejected in the previous round, will not apply to a school where this school's seats are occupied and the tentatively assigned students all have higher priority over this student(which means that the student has no choice to be admitted). Let's call it Modified DA.
These two mechanisms have exactly the same result since the modified DA just make the process more compact.
My question is: image that in step k of Modified DA, which step in the original DA, should step k in Modified DA corresponds to?
I am asking because I am doing a proof, where I know the step of the modified DA, but I need to count in which step in the Original DA. Thanks a lot for your attention!