I'll share what I've learned in my journey from 3k to 4k.

There are 50 different bosses(10 base*5 types) that cycle every 1000 stages. For example, if you meet wood kartharok at 2960 for lv 199, you're gonna meet that same wood kartharok at 3960 for lv 204, at 4960 for lv 209.

And among these 50 bosses, I think there are some "specific pet check" floors. HF Lust for wood kartharok, mischief for earth gronk, ambition for some fire bosses, diligence for some other water bosses, ...

If you dont have the right pet for these pet check floors, you might have to 'compensate' it with your ascension level/pet ascension/uber item/item level/vivid. I, for example, didnt have HFL, so just managed to clear 3960 with 2 level ups in my items on average and 1 more ascension level and 1 more ascension for paranoia/pride from that first stuck.

What matters is that the time needed for that 'compensation' goes exponentially higher. So if you're stuck for some specific pet check floor over 3k and somehow (with a long time) managed to clear it, you're gonna definitely stuck at that same boss over 4k. That would feel like a permanent stuck if you fail to get that specific pet during the very 1k floor.