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duration of hospitalizations containing both overlapped and not overlapped procedures

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I have a dataset with start (dt_ini) and end date (dt_fim) of different procedures per hospilalization (id_aih). I would like to know the duration of each hospitalization. The complexity lies in the distribution of different procedures for the same hospitalization over time, since they can either overlap or not. So I need to add the duration of different procedures that do not overlap and take the difference from the max end date to the min start date of procedures which do not overlap. I can have both kind of procedures for the same hospitalization and there is no fixed number of procedures per hospitalization. Below I give some examples of what can happen in my dataset. I am looking for a command which woud give me a final column which total numbers of day for each hospitalization (so, a column with same values for same hospitalizations).
Thank you so much.

Code:
input int id_aih    int(dt_ini    dt_fim)
1    13165    13208
1    13165    13239
1    13165    13269
1    13165    13179
2    13200    13201
2    13205    13208
2    13110    13112
3    13165    13170
3    13168    13172
3    13180    13185
4    13170    13188
4    13165    13300
4    13185    13190
4    13200    13210
end

format dt_ini dt_fim %td

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