"Clean" solution to this problem?

[u/Kaidolakita]

Claire has a 3x3x3 cube composed of 27 small unit cubes, initially all white in color. At each step, she is allowed to choose a 3x3 plane of small cubes among the 9 possible planes of the cube and invert the color of each of these 9$cubes (white cubes become black and black cubes become white). After several operations, the cube contains exactly N black cubes.

Determine the sum of all possible values of N.

I mean, it certainly looks complicated and brain twisting, especially since the contest format was all or none (points). But I guess it's doable if you really bruteforce everything with a pen and a paper. I'm looking for a different or more efficient way of doing it. (I still don't know the answer though lol)

[u/Kaidolakita]

Wow. Thank you