As vanessa worked on the problem, she reported out loud, "no, that option doesn't seem to work. no, that doesn't work either." then vanessa abruptly shouted, "i think i've got it! " these reports seem to capture the phenomenon called

The correct answer should be illumination.
The word illumination has to do with light (lumen in Latin means light), so when you suddenly come up with an idea that you couldn't previously think of, it is a sort of an enlightenment, an epiphany, which is what illumination is about. This is what happened to Vanessa in the case above.