I plugged all the cables to the nest adapter and figured that the blue wire which wasn't used on the old setup is the common wire so I plugged that in the C position for the plate and on the furnace side I added it to a pigtail that had the R wire on it. After charging it, it turns out that it cannot detect any power from the R wire but 24V is detected on the C wire. Since I wasn't making any headway I rewired the old thermostat and that works fine so I do not think I've burnt out anything. I am not sure where to go from there so any help is appreciated. Here are the pictures from the furnace side:
Somebody misplaced the blue wire from the system to the thermostat
It appears that the previous installer, for some bizarre reason, connected the blue wire and the red wire in the cable from the indoor unit to the thermostat to the fat red R wire coming from the indoor unit. Luckily, between the wiring diagram on your unit showing the fat brown wire as the indoor unit's C wire, and the fact we can trace the air conditioner control from the Y wire from the thermostat, to the red wire in a cable, with only a white wire returning to join up with that brown C wire, means that we can simply swing the blue wire from the thermostat over to join with the other wires connected to the fat brown C wire and put a replacement wirenut on the junction of the red R wires in order to get everything in order for your Nest to work.