HMMT 2014 U27

Suppose that $\left(a_{1}, \ldots, a_{20}\right)$ and $\left(b_{1}, \ldots, b_{20}\right)$ are two sequences of integers such that the sequence $\left(a_{1}, \ldots, a_{20}, b_{1}, \ldots, b_{20}\right)$ contains each of the numbers $1, \ldots, 40$ exactly once. What is the maximum possible value of the sum

$$\sum_{i=1}^{20} \sum_{j=1}^{20} \min \left(a_{i}, b_{j}\right) ?$$