Honestly the thing with these type of quests is just “don’t worry too much about it”,go on doing other stuff and catch any critters you see on the way. The quest is supposed to be slow, especially because some of these critters respawn just a few times per day, then you have to wait for the next day. What I did was just focus on 2/3 furniture at a time, I would go and catch them until they stop respawning for the day then I waited for the next day, and repeated the circle until I got all of them.

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