Meeting Room Scheduler

In this problem, you are tasked with developing a scheduling system for managing the booking of a meeting room. The system should support two primary operations:

1. book(start, end): Books a meeting from time start to time end. You can assume that start is always less than end. The times are represented as integers along a continuous timeline.

2. totalBookedTime(): Returns the total amount of time during which the room is booked. Overlapping booking intervals should be merged so that any overlapping time is only counted once.

For example, if the following bookings are made:

• book(1, 3)
• book(2, 4)
• book(5, 7)

The merged intervals would be [1, 4) and [5, 7), which results in a total booked time of (4 - 1) + (7 - 5) = 3 + 2 = 5.

Your task is to design and implement a data structure that efficiently supports these operations. Consider edge cases, such as booking intervals that partially or completely overlap existing bookings.

Please implement the data structure and its associated methods as described.