GAIC Math 2024 Problem 129

Given an integer $n > 2$. Now, there are $n$ people playing a game of "Passing Numbers". It is known that some people are friends (friendship is mutual), and each person has at least one friend. The rules of the game are as follows: each person first writes down a positive real number, and the $n$ positive real numbers written by everyone are all different; then, for each person, if he has $k$ friends, he divides the number he wrote by $k$, and tells all his friends the result obtained; finally, each person writes down the sum of all the numbers he hears. The question is: what is the minimum number of times that someone writes down different numbers?