What is reinstatement, and how long does it take?

It's asking USCIS to restore your F-1 status from inside the US. Sequence: admission at a supporting school, new I-20, then file the reinstatement. Typically 3-8 months, and you can't work or use CPT while it's pending - work resumes once status is restored. Whether it beats leaving-and-re-entering depends on your case; that's a senior-consultant conversation.