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Verb: chase after
  1. Go after with the intent to catch
    "The policeman chased after the mugger down the alley";
    - chase, trail, tag, give chase, go after, track, chase down
     
  2. Pursue someone sexually or romantically
    "He spent months chasing after his crush before finally asking her out";
    - chase

Derived forms: chased after, chases after, chasing after

Type of: court, follow, pursue, romance, solicit, woo