if you need to re-wire(i suggest you do) attach string or the 12 gauge wire(either or) to the 16 gauge you already have installed. Pull the 16 gauge through the walls along with the 12 gauge/string. If you attach string, do the same trick and attach the 12 gauge later. Just be sure you don't lose the 16 gauge in the middle of the wall and you have no wires at all.

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