It wasn't that obvious to me, though upon further reflection, I think I'm starting to see that there are a finite number of m, n pairs that can produce arbitrary integer \(\displaystyle l \in \{a, b, c\}\).

This image really helped, though. For any $l$ (at least in the range shown), it looks like you can draw a finite boundary containing triples $[a,b,c] : MIN(a,b) \leq l$ that has a relatively small number of triples within it. Even adding in the non-primitive triples isn't going to increase the number that much.