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To "connect the lines" and ensure that the average intensity is calculated for each segment including the end points that connect, I decided to generate a new row for each individual segment. It would take on one higher sequence number than the original highest sequence number per segment, and then copy in the first intensity and spatial object from the first sequence of the next segment.
Thought I had it done in simple 3-4 tool workflow until I realized there were gaps between the segments... so I created new points for each segment that corresponded to the start point of the next segment.
At first, it looked straightforward to me, but my lines had gaps between them. This happened because I treated each segments as completely separate line. Segment 4 had no line at all, as it consisted of one point only. This made me think that I need to connect each segment with the previous one. The end point of one segment should be exactly the same as the beginning of the next segment. I duplicated first rows for all the segments (except the first one), adjusted segment and sequence number and then run all the rows through my original workflow.
like @nick_ceneviva, @David-Carnes & @lminors (and most of the other folk) I took a few more steps to make sure the lines all touched by backing a copy of the last point of segment 2 back into segment 1
@ChristineB 's solution does the same, but WAY more elegantly than me :-)