
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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 100th press inclusive, what is the expected number of pairs of consecutive presses that both take you up a floor?

Present the answer in LaTex format: \boxed{Your answer}
user