Find the birthday

Malmer Pebane, Fames Jung, and Weven Dare are perfect logicians that always tell the truth. Malmer decides to pose a puzzle to his friends: he tells them that the day of his birthday is at most the number of the month of his birthday. Then Malmer announces that he will whisper the day of his birthday to Fames and the month of his birthday to Weven, and he does exactly that.

After Malmer whispers to both of them, Fames thinks a bit, then says “Weven cannot know what Malmer’s birthday is.”

After that, Weven thinks a bit, then says “Fames also cannot know what Malmer’s birthday is.”

This exchange repeats, with Fames and Weven speaking alternately and each saying the other can’t know Malmer’s birthday. However, at one point, Weven instead announces “Fames and I can now know what Malmer’s birthday is. Interestingly, that was the longest conversation like that we could have possibly had before both figuring out Malmer’s birthday.”

Find Malmer’s birthday.

From USAMTS

Note- Input your answer as MMDD. For example, if you think his birthday is the 21st of march, then put your answer as 0321.