Is my prayer valid after I caught the last rak'ah of 'Asr prayer, then stood for the second rak'ah without sitting for the first tashahhud, believing that I had caught the third rak'ah with the congregation, then performed the two prostrations of forgetfulness (sujood al-sahw) before the tasleem when I remembered this during the last tashahhud?

A latecomer to congregational prayer (masbūq) performs the prostration of forgetfulness (sujūd al-sahw) if he makes a mistake with his imam or in the part of the prayer he completes individually. The sujūd al-sahw is performed before the salām if he omits an obligatory act (wājib) or doubts the number of rakʿahs and neither side of the doubt is preponderant to him. It is performed after the salām if he adds something to his prayer or if he doubts and one side of the doubt is preponderant to him.