Interval Weight Merging

You are given a list of intervals. Each interval is represented as a triplet [start, end, weight], where start and end are integers (with start ≤ end) and weight is an integer value associated with that interval.

Write a function that merges all overlapping intervals while summing their weights. Two intervals overlap if they share at least one common point. The merged interval should have a start time equal to the minimum start among the overlapping intervals, an end time equal to the maximum end, and a weight equal to the sum of the weights of all those intervals.

For example, given the intervals:

[ [1, 3, 2], [2, 5, 3], [6, 8, 1] ]

The intervals [1, 3, 2] and [2, 5, 3] overlap and should be merged into [1, 5, 5]. The interval [6, 8, 1] remains unchanged.

Constraints:

• The input is a list of intervals in the format [start, end, weight].
• There may be multiple overlapping intervals.
• Return the merged list sorted by the start time of each interval.

Implement the solution using a programming language of your choice.