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I use 34AWG magnet wire for this - it is enamel insulated, so you can strip the ends by burning off the enamel with solder instead of mechanically stripping it. For soldering my favorite is a Metcal SMD rework iron with a fine conical tip, although I’ve done work like this with a lot of different irons. The Metcal is fairly expensive but something cheaper like the TS100 would be capable of doing it. The most important tool is a stereo microscope. This scale of work is just possible without magnification, but it is enormously easier under a microscope, and stereo vision is a necessity. The other thing that’s very helpful is flux - when I’m doing a solder joint to one of the pads on the microcontroller, it’s tough to add solder without adding too much, so I usually tin everything, then put a tiny amount of flux on the wire, then just reflow it without additional solder.