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Synonyms for Undertake (same or very similar meaning)

WordNet sense 1 (accept as a challenge):
tackle, take on

WordNet sense 2 (accept as a charge):
take in charge

WordNet sense 3 (enter upon an activity or enterprise):
attempt, set about

WordNet sense 4 (promise to do or accomplish):

WordNet sense 5 (enter into a contractual arrangement):
contract, contract

From the ODE community, based on WordNetadd/edit