number - Will the mesiah build the 3rd temple?

Some sites claim that the messiah will build the 3rd temple. Another says that the next temple, will, obviously be the 3rd.

However, Ezekiel simply envision an eternal temple. He never said it'll be the 3rd. Ezekiel lived during second temple era. Did he foresee that the 2nd won't make it till the end of time?

Why 3rd?

As opposed to the 4th, 5th, 6th, 100th? How do we know? Anything on tanach telling that?

Answer

Rashi (Sukkah 41a "I nami") says that the third temple will come down built from heaven.