Is the action of an Imam who prostrated once in the first rak'ah, then performed a rak'ah after the end of the prayer, prostrated, and said the taslim considered valid?

The prayer of an Imam who mistakenly omits a prostration of forgetfulness (Sajdat al-Sahw) and does not return to it, then offers an additional rak'ah shortly after the salām, is valid, and he must perform the prostration of forgetfulness. As for the congregants (ma'mūmūn), they must alert him. If he does not return, they should perform the prostration he missed and then rejoin him.