Is my prayer valid, knowing that I caught the last rak'ah of Dhuhr prayer with the Imam, and I forgot the first Tashahhud in the second rak'ah, then I remembered and sat for it in the third rak'ah, and I completed the prayer and performed the prostration of forgetfulness (sujūd al-sahw)?

Most scholars hold that the first tashahhud (middle tashahhud) is a Sunnah, and its omission is compensated for by the prostration of forgetfulness (sujud al-sahw). A latecomer (masbuq) who missed three rak'ahs in a four-rak'ah prayer should stand up and perform one rak'ah, then say the tashahhud, then perform two more rak'ahs, in order to preserve the order of the prayer. This is because performing the first tashahhud after completing another rak'ah does not alter the form of the prayer. This is the view held by Muhammad ibn al-Hasan from the Hanafis, and it is the preferred view among the majority of scholars. Some Hanafis, however, held that if one missed three rak'ahs, they should pray two, then sit, then pray the third. Therefore, it is safer to repeat the prayer because you added an extra sitting in an incorrect place.