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For each sufficiently large N, there exists a unary regular language L such that both L and its complement Lc 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 Lc 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.