The ruling on leaving off the takbir at the end of the tawaf and the sa'y

Question:
May Allah reward you. This questioner says: I performed an 'Umrah and did not say the takbir at the end of the tawaf, and I also did not say the takbir at the end of the sa'y, knowing that I was ignorant of this. May Allah reward you.

Answer:
This does not harm, because the takbir is a recommended practice (sunnah). It does not harm if you leave it off, even if you did not start with the takbir at the beginning of the tawaf and the sa'y. If you did not start with the takbir in the tawaf, it is valid. Yes.