The words act and action both mean “a deed” and “the process of doing.” However, other senses of act, such as “a decision made by a legislative body,” and of action, such as “habitual or vigorous activity,” show that act tends to refer to a deed while action tends to refer to the process of doing. Thus people commit sex acts every day, but never sex actions.

If they are seen, they are caught in the act, but not in the action. By the same token, they may want a piece of the action, but not of the act. As you can see, the demands of meaning or idiom often require one word or the other. But in cases where either can be used, either is acceptable: My act (or action) was premature