What kind of zoom lense for bird photography?

Ok so i am pretty new to photography but i am getting better. I am mostly interested in bird photography. I have taken many pictures of birds in my backyard from hummingbirds to pileated woodpeckers. The crows are too scared lol
So i am looking for a lense with good zoom, at least 400mm. My current zoom lense is 55-200mm.

So my camera is a nikon d3300. I need a lense with at least 400mm zoom for photographing wild birds. I would like to photograph other animals too but i currently dont have the opportunity to. The lense cant be very expensive because i am a teenager and unless it is under $100 (ebay?) my parents will choose whether they want to pay it or not.

I put a picture i took of a hummingbird. The picture is compressed so it is blurry when you zoom into it. But the actual picture is very good.

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