Well, after some time I was finally able to get the core replaced and placed back in. While doing so, I came across a cable that I noticed had been disconnected already. Today, as I'm working on putting the dash back in I find another cable with corresponding colors and the same amount of female prong connectors that match with the aforementioned disconnected cable. Do these cables connect to each other? (Red outlined cable is the disconnected one that leads into the A/C Unit Housing, Blue outlined cable is the one I found in the dash and seems to have a cable plugged in to complete the circuit.)