Order the passes by cost of failure
Edit in the wrong order and you re-do work. Fix the claims first, then the structure, then the sentences, then the words. Each pass depends on the one above it being settled.
Editing prose before verifying claims produces a beautifully written inaccuracy. Restructuring after line editing means throwing away the lines. The cheap sequence is always facts, then shape, then language.
The automated checker belongs at the start of that sequence, because most of what it finds is cheap to fix and expensive to discover later.