Weird coolant leakage

Riddle me this:

This spring when the warm temperature came, I first found that my AC had given up the gist.

Then I started finding puddles of coolant below the car. They were random and sporadic - nothing for days, then suddenly a small lake. I searched inside the engine compartment but found no obvious culprit. Bought a large can of coolant and topped up now and then.

Then I figured I should refill the AC, and bought a $20 can of R1234something and topped it up a bit. AC started working again, albeit not at full strength.

And the coolant leaks disappeared. Almost. What used to be a liter per week turned into a liter since summer.

I'm an engineer with what I like to delude myself is a somewhat good understanding about how machines work, but this is beyond me. What does the AC have to do with the engine cooling system - and vice versa - such that this occurs?

E500, M273