Maximize $\sum x_{k}^{2}\sum kx_{k} $ for $x_1 \ge \cdots \ge x_n \ge 0$ and $\sum x_k = 1$

Let $ x_{1}\geq x_2\geq\cdots\geq x_{n}\geq0$, and $\sum\limits_{k=1}^{n}x_{k}=1$. What is the maximal value of$$\sum_{k=1}^{n}x_{k}^{2}\sum_{k=1}^{n}kx_{k} ?$$

I think, Maybe we could try Rearrangement inequality

or

let $x_n=a_n,x_{n-1}=a_n+a_{n-1} , \dotsc, x_1=a_1+a_2+\dotsc+a_n$

where $a_i\geq0$

I am really grateful for any help

EDIT: I am very sorry, $1$ is not the upper bound for all $n$, but I guess $\sum\limits_{k=1}^{n}x_{k}^{2}\sum\limits_{k=1}^{n}kx_{k}\lt2 .$

For $n=5$$$x_1=0.926599, x_2=x_3=x_4=x_5=0.0183503$$LHS $=1.01773\gt 1$