Omni-MATH / Two rational numbers (m)/(n) and (n)/(m) are written on a blackboard, where m and n are relatively prime positive integers. At any point, Evan may pick two of the numbers x and y w…

Two rational numbers (m)/(n) and (n)/(m) are written on a blackboard, where m and n are relatively prime positive integers. At any point, Evan may pick two of the numbers x and y written on the board and write either their arithmetic mean (x+y)/(2) or their harmonic mean (2xy)/(x+y) on the board as well. Find all pairs (m,n) such that Evan can write 1 on the board in finitely many steps.

Two rational numbers $\frac{m}{n}$ and $\frac{n}{m}$ are written on a blackboard, where $m$ and $n$ are relatively prime positive integers. At any point, Evan may pick two of the numbers $x$ and $y$ written on the board and write either their arithmetic mean $\frac{x+y}{2}$ or their harmonic mean $\frac{2xy}{x+y}$ on the board as well. Find all pairs $(m,n)$ such that Evan can write $1$ on the board in finitely many steps.