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We also derive a new interpolated Harnack inequality for the equation $\\omega_t=\\Delta\\omega-\\omega\\ln\\omega+\\varepsilon R\\omega$ on closed surfaces under the $\\varepsilon$-Ricci flow. Finally we prove a new differential Harnack inequality for the equation $\\omega_t=\\Delta\\omega-\\omega\\ln\\omega$ under the Ricci flow without any curvature condition. 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