Pat and Nat are dating, and all of their dates are scheduled to start at $9$pm$.$ Nat always arrives promptly at $9$pm$.$ Pat is highly disorganized and arrives at a time that is uniformly distributed between $8$pm and $10$pm$.$ Let X be the time in hours between $8$pm and the time when Pat arrives. If Pat arrives before $9$pm$,$ their date will last exactly $3$ hours. If Pat arrives after $9$pm$,$ their date will last for a time that is uniformly distributed between $0$ and $3$ X hours. The date starts at the time they meet. Nat gets irritated when Pat is late and will end the relationship after the second date on which Pat is late by more than $45$ minutes. All dates are independent of any other dates.

$($a$)$ What is the expected number of hours Nat waits for Pat to arrive?

$($b$)$ What is the expected duration of any particular date?

$($c$)$ What is the expected number of dates they will have before breaking up$?$