My favourite solution to this problem was going all in on analyzing the input. Instead of just assuming the set of modules must have cyclic behaviour and running the simulation until you find the periods, look at the input and _really_ understand what its doing.

What you will find is that the modules form a set of binary counters (chains of flip flops), with a number encoded into them via whether they are connected to a "hub" node of the chain (a conjunction).

You can parse the module structure and traverse the graph to extract that number. The connections to the hub are the bits of the number (1 if module is connected, 0 if not). Do that for all the counters and LCM (or multiply since they are all coincidentally co-prime) them together to get your answer.

No simulation required.