Just as a followup to underline the need for caution and diligence by OW fliers.
The agent promise to reinstate and call me that I received at 7PM at night had not materialized by 9:30AM the next day, with my next segment at 1:15PM now getting perilously close.
I used the hotel concierge to call. The LAN status had relapsed to "we cancelled your flights because you or a prior airline failed to .. . ". Now having had two agents tell me things were okay, when they were not, my concern level was rising.
I armed the concierge with the important points, most key being the phrase "involuntary reroute by LAN", wiht the flight numbers and dates involved. The concierge at the JW Marriott has a good relationship with someone in LAN, and after a bit, they finally did (again) admit that it was their error to cancel the segments. They promised to rebook all the segments in 20 minutes, which the concierge would verify by calling back after that interval.
It only took 45 minutes, but the concierge called back to tell me all were now rebooked, but that I should be heading immediately to the airport due to my near-in flight.
I did log in to lan.com and select seats, as the rebooking was done but nothign beyond that.
In fact, when I got to the airport, they had reserved the seats but having already deleted all teh eticket and fare info, I had a reservation with no sign of payment. Fortunately I had collected itineraries and receipts every way I could, so that I did have somewhere paper ticket stock receipts that listed the eticket numbers. Took 45 minutes but finally we had boarding passes.
Same problem again in Cusco coming back, but the agent took the extra step of printing the eticket numbers on our boarding passes.
While the error was fixed, it was in the most minimal and incomplete way. Be vigilant and have lots of backup documentation.