If you haven't seen this problem yet, do try it out. **Spoilers follow.**

I would like to propose an extension: is it possible to improve the bound \(O(n!^3)\)?

An idea I had is start with a primorial cubed, multiply by 8 each step, until the next primorial cubed is needed. Of course, this would allow arbitrarily large subsequences of geometric sequences, so we will not allow increments of 8, ie \(\dfrac{t_{n+1}}{t_n}\) is strictly monotonically increasing.

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