Short Answer
Your work probably has to be released under the GPL and you probably won't be able to commercialize it.
Long Answer
First, let's get the MIT- and BSD-licensed dependencies out of the way. Those don't impose any restrictions on how you license your work, but they do have other requirements attached (such as attribution). I'm assuming those requirements aren't a problem for you.
Now, the GPL. The big question is whether your work is "based on" a GPL-licensed work to such a degree that it is legally considered derived from it. If you're just duplicating functionality or taking inspiration from the user interface, then it's probably not derived, and you can license your project however you want. On the other hand, if you're forking the original project's source code, your project is almost certainly derived.
If your work is derived from the original work, then you need to release your work under the GPL too. After all, that's the whole point of copyleft.
Can a GPL work be commercialized?
This depends on your business model. The GPL doesn't forbid you from selling your software – actually,
the FSF encourages charging distribution fees for fundraising – but it does make earning a profit very difficult. You will have to provide corresponding source alongside every binary distribution, and anyone who has the software will be able to redistribute it to anyone else. If you think you can make that work, go for it.
Another Option
You could always try negotiating directly with whoever owns the copyright to the GPL work you want to derive from. If they themselves are not bound by GPL-licensed dependencies, they might license their work out to you under different terms.
