Suppose we changed the definition of a non-deterministic finite state automaton so that it accepts a string w if and only if not

Suppose we changed the definition of a non-deterministic finite state automaton so that it accepts a string w ∈ Σ∗ if and only if not only there exists a processing path of w that consumes all symbols of w and arrives at an accepting state, but every all-symbol consuming path of w arrives at an accepting state. Show that the family of languages accepted by such machines is precisely the set of regular languages.