3 Pickup Guitar, Rythm switch not working

I recently bought my first electric guitar. I have played an acoustic fender dread for about a year and a half. I have no experience with the electronics of guitars but I am trying to learn. I want to fix this problem myself, and hence learn enough about my guitar to set it up/mod it/fix it properly myself but need some help doing so. I am a graduate student so cost is a factor (I’d like to do this all on the cheap)

I bought the guitar off craigslist for $150. I believe it is a 2007 Epiphone Les Paul Custom, Black Beauty Edition. I do not know if the electronics are stock but the prior owner said they were.

TLDR:
The guitar has 3 pickups. The rhythm switch of the 3 way switch does not have any output. I do not know which pickup is turned on from each position on the switch-whether the middle is wired to be all 3 pickups or the middle is incorporated in the up or down switch (I know the neck pickup is the rhythm switch on a traditional 2 pickup setup).

I want to fix this and am willing to buy new pickups but don’t want to invest that type of money if this is an easier fix. How do I diagnose whether this is an issue with the pickup, or wiring or 3-way-switch? Also any opinion as to upgrading the pickups?

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