Interval Merging Scheduler

You are tasked with implementing an event scheduler that manages time intervals for events. Each event is represented as an interval [start, end] where start and end are integers and start ≤ end.

Your scheduler should support the following operations:

1. Merge Intervals: Given an unsorted list of intervals, merge all overlapping intervals and return a list of non-overlapping intervals that cover all the time spans of the original list.

   • Example:
     • Input: [[1, 3], [2, 6], [8, 10], [15, 18]]
     • Output: [[1, 6], [8, 10], [15, 18]]

2. Add Interval: Insert a new interval into the scheduler, merging it with any existing overlapping intervals so that the scheduler continues to maintain non-overlapping intervals.

3. Remove Interval: Remove an interval from the scheduler. This may involve splitting an existing interval into two if the removed interval is in the middle of an existing interval. The removal should ensure that the remaining intervals accurately represent the remaining scheduled time.

Implement these operations in your preferred programming language with an emphasis on clean code and efficiency. Make sure to consider edge cases such as:

• Inserting an interval that is completely separate from existing intervals.
• Adding an interval that overlaps with multiple intervals.
• Removing an interval that partially overlaps with an existing interval or extends across multiple intervals.

Write a complete solution that demonstrates all functionalities of the scheduler.