Cannot set up the old account in a reinstalled Microsoft Authenticator

Question

Recently, I accidentally uninstalled Microsoft Authenticator from my phone. After reinstalling it, the app brought up the previously configured accounts, but asked me to log in to each one again. When I enter my password, it asks me to enter a code provided in the Authenticator itself (which doesn't exist, as I'm trying to set up this account in the app). When I ask to use another login method, it only offers me two options, and both require confirmation with the Authenticator itself. How can I escape this endless loop?