INMO 2025 Problem 2

Let $n\ge 2$ be a positive integer. The integers $1,2,\cdots,n$ are written on a board. In a move, Alice can pick two integers written on the board $a\neq b$ such that $a+b$ is an even number, erase both $a$ and $b$ from the board and write the number $\frac{a+b}{2}$ on the board instead. Find all $n$ for which Alice can make a sequence of moves so that she ends up with only one number remaining on the board.Note. When $n=3$ , Alice changes $(1,2,3)$ to $(2,2)$ and can't make any further moves.

Proposed by Rohan Goyal