```python
def merge_intervals(intervals: list[list[int]]) -> list[list[int]]:
    """
    Merge overlapping or touching intervals in a list of intervals.
    
    :param intervals: List of intervals where each interval is a list of two integers [start, end].
    :return: A new list of merged intervals sorted by start.
    """
    if not intervals:
        return []
    
    # Sort intervals based on the start value
    intervals.sort(key=lambda x: x[0])
    
    merged = [intervals[0]]
    
    for interval in intervals[1:]:
        last_merged = merged[-1]
        
        # Check if the current interval overlaps or touches with the last merged interval
        if interval[0] <= last_merged[1]:
            # Merge the intervals
            merged[-1][1] = max(last_merged[1], interval[1])
        else:
            # Add the new interval to the merged list
            merged.append(interval)
    
    return merged
```