OFM 2023 J2

On her blackboard, Alice has written $n$ integers strictly greater than $1$ . Then, she can, as often as she likes, erase two numbers $a$ and $b$ such that $a \neq b$ , and replace them with $q$ and $q^2$ , where $q$ is the product of the prime factors of $ab$ (each prime factor is counted only once). For instance, if Alice erases the numbers $4$ and $6$ , the prime factors of $ab = 2^3 \times 3$ and $2$ and $3$ , and Alice writes $q = 6$ and $q^2 =36$ . Prove that, after some time, and whatever Alice's strategy is, the list of numbers written on the blackboard will never change anymore.

Note: The order of the numbers of the list is not important.