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...yes, I know; mouthful of a title, but it gets the premise out fast.
Two events happened in OTL October 1912, within weeks of each other:
October 14: Former President and then-current candidate Theodore Roosevelt is shot by John Schrank before giving a speech in Milwaukee; he recovers.
October 30: Vice President James S. Sherman succumbs to Bright's disease.
The butterfly is this: ITTL, in the week in between those two events, President William Howard Taft suffers a fatal heart attack.
What happens?