As written in Japanese editorial, any state to consider can be expressed in a form $$$\lfloor N / (2^p 3^q 5^r) \rfloor$$$, where $$$p, q, r$$$ are non-negative integers, since $$$\lfloor \lfloor N / x \rfloor / y \rfloor = \lfloor N / xy \rfloor$$$. Each of $$$p, q, r$$$ have only $$$O(\log N)$$$ possibilities, so in total the number of states is $$$O(\log^3 N)$$$.

I think "picks up" means that he owns the accelerator. That is, if he visit a treasure box which is already visited, there is no accelerator because he has it. For the statement Each time he picks up an accelerator, his moving speed gets multiplied by 2., i think the writer wanted to say that if he has $$$k$$$ accelerator, his current speed is $$$2^k$$$.

Therefore, we can see the state that some treasure boxes are visited multiple times as those treasure boxes are visited once.

Ah I see. First of all, each treasure box contains only one booster. That is, even if we visit the same treasure box again and again, the multiplier from that booster is just 2.

"in the same way by ~" means that we also maintain which treasure boxes is already visited.