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Jelly, 16 bytes

’ÆRṡ2_/2+$$ÐḟFİS

Try it online!

While trying to improve my previous answer, I thought up a totally different algorithm, and it comes in somewhat shorter. I'm using a different post for it, as is the standard here for an answer that uses a different technique.

(EDIT: Deleting, it's too close to Dennis' answer which was posted while I was working on this.)