How would I fix this linear tracking turntable solenoid?

https://imgur.com/a/APCtxlP

List of images of the parts and section of particular parts in the service manual above.

Hey all, so I'm trying to repair a Sony PS-FL7 turntable that was long neglected and seemed to be pretty well knocked around in shipping. Most of the main issues I've seemed to be able to troubleshoot and the linear tracking seems to be operating smoothly, as well as the front loading mechanism. The last thing I need to fix thought before I think I'll be able to actually do some testing of the sound output is the tonearm lifter.

When I opened the whole turntable up I found the arm drive coil jsut laying in the empty space under the platter and it had obviously come undone from it's board and was making it impossible for the tonearm to drop. My assumption is that once I get this solenoid working again that I'll be up and running, however this is a good bit beyond my knowledge, as I understand the principle for how a solenoid should work, but rarely do any actual electronics repair like this.

From looking at the service manual, it appears that the coil is supposed to be wired to the leads on the board that it was originally glued to, however there aren't any obvious wires to solder there. When messing around a bit I was able to sort of parse out what seems to be one tiny strand of copper wire that's coming off of the bottom of the coil, and under the old glue maaaaaybe what looks like another line that was cut off right at the edge and therefore doesn't stick out at all. Is it likely that these are the connections that would have been connected to the board's circuits? Would I be able to use these somehow to reconnect it? I thought the construction of a solenoid requires a complete circuit, but if it's just a bundle of copper wire, does it really matter which part is connected to which lead?

I guess I'm open to all advice as to how to best repair this. I'd rather not try to track down a whole other replacement tonearm lifter if I have everything in hand to make it work, so anything is helpful.

Last photos is just kind of to get a sense of the way it's assembled. It's all undone and loose and probably not necessary to understand what's going on, but figured it might be useful.

Thanks!