take off after
phrasal verb
            
              took off after; taken off after; taking off after; takes off after            
        
    
        US      
      
                                
                                
              
          
                                                      : to start to move quickly in order to catch (someone or something)                                      
              
                             
He jumped in his car and took off after them.
                         
                
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  Merriam-Webster unabridged




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