10. Given a positive integer $n(n \geqslant 2)$. It is known that $2 n$ positive real numbers $a_{1}, a_{2}, \cdots, a_{2 n}$, satisfy:
$$
\sum_{k=1}^{n} a_{2 k-1} \cdot \sum_{k=1}^{n} a_{2 k}=\prod_{k=1}^{n} a_{2 k-1}+\prod_{k=1}^{n} a_{2 k} .
$$

Find the minimum value of $S=\sum_{k=1}^{2 n} \frac{a_{k}^{n-1}}{a_{k+1}}$, where $a_{2 n+1}=a_{1}$.