Pairs of Complementary Unary Languages with "Balanced" Nondeterministic Automata

For each sufficiently large N, there exists a unary regular language L such that both L and its complement L^c are accepted by unambiguous nondeterministic automata with at most N states while the smallest deterministic automata for these two languages require a superpolynomial number of states, at least $e^{\Omega(\sqrt[3]{N\cdot\ln^{2}\!N})}$. Actually, L and L^c are accepted by nondeterministic machines sharing the same transition graph, differing only in the distribution of their final states. As a consequence, the gap between the sizes of unary unambiguous self-verifying automata and deterministic automata is also superpolynomial.