Verb chase after has 2 senses
  1. chase, chase after - pursue someone sexually or romantically
    --1 is one way to woo, court, romance, solicit
    Sample sentence:
    Sam cannot chase after Sue
  2. chase, chase after, trail, tail, tag, give chase, dog, go after, track - go after with the intent to catch; "The policeman chased the mugger down the alley"; "the dog chased the rabbit"
    --2 is one way to pursue, follow
    Sample sentence:
    They chase after the car down the avenue
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