Let $x_1,x_2,...,x_k$ be real numbers such that $\sum_{i=1}^k x^2_i=1$. Determine the minimum and maximum (if there is) value of $$\prod_{i=1}^k (x_i-x_{i+1})$$ and determine all values of $(x_1,x_2,x_3,...,x_k)$ such that the minimum value is achieved. ($ k \ge 2$)
This question is a generalized version of the question APMO 2022 P5 ($k=4$), and regional stage of All-Russian MO 2018, 9.5. ($k=3$).
The problem is, since this is a generalized version, the strategy I did in $k=3,4$ does not work here. In small $k$, it is sufficient to use AM-GM, but since there are more variables in this question, I have no idea how do attack this problem, can someone help me please? I don't need solution, just a hint to solve the generalized version