After doing the free quizzes about formal language, I got one question to be answered.
............I look forward to hearing from you.
1/ Thank you
2/ Thanking in advance
The answer is the second one, Thanking in advance.
Why not Thank you in advance or Thanks in advance?
I don't know; it may be a typo. The sentence structure you have presented does not allow of either answer.
In any case, I think the -ing form should be avoided as effete.