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Wiktionary
go after

vb. 1 To pursue in attempt to catch another. 2 To pursue an object or a goal.

WordNet
go after
  1. v. go after with the intent to catch; "The policeman chased the mugger down the alley"; "the dog chased the rabbit" [syn: chase, chase after, trail, tail, tag, give chase, dog, track]

  2. go in search of or hunt for; "pursue a hobby" [syn: quest for, quest after, pursue]

Usage examples of "go after".

Then, after a period during which anyone should have become convinced of their intention to remain at the Shack forever, and never stir an inch from the place, then the pair would sneak out, and go after Chris Bede.