For any finite non-empty set $X$ of integers, let $\max(X)$ denote the largest element of $X$ and $|X|$ denote the number of elements in $X$. If $N$ is the number of ordered pairs $(A, B)$ of finite non-empty sets of positive integers, such that
\[
\begin{aligned}
& \max(A) \times |B| = 12, \\
& |A| \times \max(B) = 11
\end{aligned}
\]
and $N$ can be written as $100a + b$ where $a, b$ are positive integers less than 100, find $a + b$.

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