HMMT 2019 U20

On floor 0 of a weird-looking building, you enter an elevator that only has one button. You press the button twice and end up on floor 1. Thereafter, every time you press the button, you go up by one floor with probability $\frac{X}{Y}$, where $X$ is your current floor, and $Y$ is the total number of times you have pressed the button thus far (not including the current one); otherwise, the elevator does nothing.

Between the third and the $100^{\text {th }}$ press inclusive, what is the expected number of pairs of consecutive presses that both take you up a floor?