What should I do if I enter the mosque and find the worshippers have already begun the prayer, specifying how to act based on the number of missed rak'ahs?

If you enter the mosque and find the congregants at any part of the prayer, join them. Whether you catch the bowing (rukūʿ), prostration (sujūd), or the tashahhud, when the Imam finishes the prayer, complete what you missed, based on the Prophet’s (peace and blessings be upon him) saying: "Whatever you catch, pray, and whatever you miss, complete." And whoever catches the bowing with the Imam has caught the rakʿah.