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The incident occurred early on Saturday, at Haast.
After the initial incident the man threw the tomahawk over some trees and hit a nearby house occupied by a couple with a young child. No one was injured.
The man is described as European, with short dark hair and a scruffy beard. He has a New Zealand accent and he was wearing blue rugby jersey denoting a Gore team.
Any information can be sent to Snr Constable Paul Gurney at Haast on firstname.lastname@example.org or via crimestoppers - 0800555111.