What is the ruling on someone who entered the mosque while the congregants were in the last rak'ah of the Maghrib prayer, and he made the takbirat al-ihram and prostrated before the Imam, then after the Imam's taslim, he performed two rak'ahs? And is the first rak'ah invalid?

For a worshiper (ma'mum) to catch a unit of prayer (rak'ah), it is stipulated that he must join the Imam in the bowing (ruku') before the Imam rises, and that he performs the minimum valid bowing with the Imam, which is bending such that his hands can reach his knees. Imam Al-Shafi'i and his companions have explicitly stated this. The author of "Al-Bayan" stipulated that the worshiper must achieve طمأنينة (stillness/composure) in the bowing before the Imam rises. There is another opinion that does not stipulate the worshiper joining the Imam for the duration of saying "Subhanallah," but rather it is sufficient for them to be together in the minimum valid bowing. Based on this, if a person does not catch the bowing with the Imam, his prayer is invalid and he must repeat it.