Is it obligatory for the congregant to perform the prostrations of forgetfulness with the Imam if the Imam made a mistake and the congregant did not make a mistake in performing the rak'ahs?

The correct [ruling] is to perform the prostration of forgetfulness (sujud al-sahw) if the Imam forgets, because if the Imam forgets, the follower (ma'mum) is bound by the ruling of his forgetfulness. You should have prostrated when the Imam alerted you while you were in the mosque. If you exited the mosque and did not prostrate, then there is no prostration required of you after that, and your prayer is valid. Some scholars are of the opinion that it is permissible to perform the aforementioned prostration after leaving the mosque, and after a long time has passed since the prayer, even if it is after a year, and this is the madhhab of the Malikis. Accordingly, it is legislated for you to perform the prostration of forgetfulness as long as you are in the mosque, and outside it according to some scholars, and your prayer is valid.