One, (40 points) Given $\sum_{i=1}^{n} a_{i} \leqslant n, \sum_{j=1}^{n} b_{j} \leqslant n$. Find
$$
\sqrt{2} \sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\frac{1}{a_{i}}+\frac{1}{b_{j}}}-\sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\frac{a_{i}^{2}+b_{j}^{2}}{a_{i}^{2} b_{j}+a_{i} b_{j}^{2}}}
$$

the minimum value.