Is my prayer valid if I prayed Maghrib alone, and after the first rak'ah, I sat for the first tashahhud, thinking I was in the second rak'ah, then I remembered that I had only prayed one rak'ah, so I made the rak'ah I was in the second and sat in it for the first tashahhud, then I stood for the third rak'ah and completed the prayer, and performed the prostration of forgetfulness before giving the salutation?

Prostration of forgetfulness (sujūd as-sahw) is correct for a worshiper who unintentionally adds something to their prayer that is of its own kind. Two prostrations of forgetfulness are obligatory to rectify the error. Ibn Qudamah stated in "Al-Mughni": "If one adds to the prayer something of its own kind, which if done intentionally would invalidate it, then prostration (of forgetfulness) is obligatory if it was done unintentionally." As for the timing of the prostration of forgetfulness, the Shafi'i and Hanbali schools hold that it is before the salutation (taslīm), while Ibn Taymiyyah preferred it to be after the salutation. In any case, the prayer remains valid.