The sentence Je regrette _ mon action needs an adverb to modify the verb regretter (to regret), expressing how intensely the speaker regrets the action.
The correct word is vivement, an adverb meaning "deeply" or "keenly." The expression je regrette vivement is a standard French collocation meaning "I deeply regret." It fits naturally between the verb and its object.
The other options do not work here:
brûlante (burning) is a feminine adjective, not an adverb. It cannot modify the verb regretter.
curieusement (curiously) is an adverb, but saying "I curiously regret my action" is semantically nonsensical.
dure (hard/harsh) is an adjective in its feminine form. While dur can occasionally function as an adverb in informal French (travailler dur), dure (feminine) cannot, and regretter dur is not a valid expression.
When expressing intensity with regretter, the standard adverbs are vivement, profondément, or sincèrement.
