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More Olympic Ring Addition

Given the five olympic rings how can the digits one through nine be placed
within the nine regions (five non-overlapping ring regions and four overlapping
regions shared between two rings) so that each ring contains a successively
incrementing total?

Or alternatively, using the labels A through I for the regions, how
can the numbers one through nine be assigned to the variables such
that:

A+B = B+C+D-1 = D+E+F-2 = F+G+H-3 = H+I-4
How many different totals can fit in the first circle? Find a complete
solution for each.

Source: Original.
Similar to previous POTW July 25, 1997

Solution

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