Let $a_1, a_2, a_3, \ldots$ be a non-decreasing sequence of positive integers. For $m \ge 1$, define $b_m = \min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n \ge m$. If $a_{19} = 85$, determine the maximum value of $a_1 + a_2 + \cdots + a_{19} + b_1 + b_2 + \cdots + b_{85}$.

Let's think step by step and output the final answer within \boxed{}.
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