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The exclusive reactions $\gamma p \to \bar K^0 K^+ n$ and $\gamma p \to \bar K^0 K^0 p$ have been studied in the photon energy range 1.6--3.8 GeV, searching for evidence of the exotic baryon $\Theta^+(1540)$ in the decays $\Theta^+\to nK^+$ and $\Theta^+\to p K^0$. Data were collected with the …