Recently, I was watching a video by an authentic Vaishnava guru named Sri Dharma Pravartaka Acharya (who has a YouTube channel called DharmaNation). He was doing a live Q&A session and touched upon the topic of the Hare Krishna maha mantra. Now, although he is a Shri Vaishnava, he also had a guru who was a Gaudiya Vaishnava, so he studied both the Chaitanya Charitamrita and the Chaitanya Bhagavata.
Now, he said that Chaitanya Mahaprabhu did indeed espouse chanting Hari's names, but interestingly, in neither work does one find the explicit mention of the Hare Krishna mantra. He goes on to say that if Chaitanya Mahaprabhu did chant this mantra, we certainly do not know about it in written records.
Is this true? Is there any mention of the Hare Krishna maha mantra in either the Chaitanya Charitamrita or the Chaitanya Bhagavata? Other legitimate sources are fine as well. I am looking for both direct and indirect references if possible.
If not, then how exactly did the maha-mantra get into Gaudiya Vaishnavism?