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Rangehood needs ducting to outside wall or through roof

I had a rangehood installed, however it's not ducted to the outside. I need someone who understand the healthy homes standards to ensure this meets the needs for ventilation. I'm looking to keep the existing range hood, just install the ducting and conceal the ducting to keep a tidy appearance. As you can see in some images, the cornice needs replacing. Needs this done to a high standard, not a rush/cheap job. Roof is flat, so wall may be best. Thanks.