MySQL 5.7 via brew on MacOS 12.6: how to get past "Access denied" barrier?

I've been through every how-to I can find on the internet, but none are working for me. I've got a fresh install of MySQL 5.7 via homebrew. I can see that the service is running, and the mysql client is installed as well. Any attempt to get to the database via the client, like mysql -u root -p or more simply mysql -u root as the installation process suggests, asks for a password with the prompt "Enter password:". Well, I just installed the database; I have no idea what the password is. Using sudo makes no difference. I've tried solutions involving mysqladmin with essentially the same result: "Access denied for user 'root'@'localhost'". I've got a bunch of experience with Postgresql, but it's got a significantly different way of dealing with mapping platform identity to db identity apparently. How do I get this thing to work? (Note that the many answers in Ask Different are among the ones I've tried already with the exact same problem.)