13. Given $a_{1}, a_{2}, \cdots, a_{n}$ are positive real numbers, and $a_{1}+a_{2}+\cdots+a_{n}=1, k$ is a positive integer, find the minimum value of $\frac{1}{a_{1}^{k}\left(1+a_{1}\right)}+\frac{1}{a_{2}^{k}\left(1+a_{2}\right)}+\cdots+\frac{1}{a_{n}^{k}\left(1+a_{n}\right)}$.