Plz help me, connect a common wire for smart thermostat in furnace

I bought a 3rd generation Nest. It doesn’t turn on after I hooked it up. I did research saying I need a common wire as power supply.

I have closely look all the wires in my system and drew a diagram. Can someone take a look see if my solution going to work? There is a spare wire in place already, it’s just not connected on either end(in thermostat and in the furnace)

There are two common wires inside the furnace, one return itself, the other connects to the condenser (this is the one I am thinking connect to the thermostat).