BALKAN 2026 Problem 4

Let $n \geq 2$ be an integer. Initially, the number 1 is written $n$ times on a blackboard. An operation consists of choosing two numbers $a$ and $b$ currently on the blackboard, not both zero, and replacing them with the numbers $$\frac{(a - b)^2}{a + b} \quad \text{and} \quad \frac{4ab}{a + b}.$$ Determine all integers $n$ for which it is possible, after a finite number of operations, for the number $n$ to appear on the blackboard.