There is no problem using "his" to refer to "Racine's" because "his" is possessive as well. What the GMAT doesn't like is the use of anon- possessive pronoun to refer to a possessive noun. So we can't say "Racine's life in the theatre prepared him well for politics." This isn't tested often, and don't expect to see it followed much in the real world. It's kind of a silly rule.
Note that it's okay to have a possessive pronoun refer to a non-possessive noun.
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Note that it's okay to have a possessive pronoun refer to a non-possessive noun.
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