Is my prayer valid, knowing that I forgot a prostration in the first rakʿah, then performed it intentionally in the second rakʿah, then performed a third rakʿah in lieu of the one in which I did not prostrate, and then performed the two prostrations of forgetfulness (sujūd al-sahw)?

If someone forgets a pillar (such as prostration) and it is possible to make it up, they should perform it and whatever follows it. If it is not possible to make it up, they should nullify that rak'ah and perform a substitute for it, and then perform the prostration of forgetfulness (sujud al-sahw). The opportunity to make up a pillar is lost upon beginning the recitation of the next rak'ah. In such a case, the rak'ah in which the pillar was forgotten is nullified, and the subsequent rak'ah takes its place. Therefore, if you remembered the forgotten prostration after beginning the recitation of the second rak'ah, your prayer is valid.