I would hope that whoever sold you the crossbow made you aware that there are minimum requirements that need to be met before it can be used on DOC land
Bow and cross bow hunting: Permits and licences
Here's the page with info about getting your permit. It's as simple as filling in the online form and hit send. Permit comes in an email straight away.
Hunting permit: Permits and licences
I've read that Matawai conservation area (6 hrs drive from AKL) has a few goats.
The cullers are going through Pirongia (2hrs from AKL) at the moment according to DOC website.
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