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We consider a three-mode optomechanical system where two cavity modes are coupled to a common mechanical oscillator.We focus on the resolved sideband limit and illustrate the relation between the significant parameters of the system and the instantaneous-state mean phonon number of the oscillator cooled to the ground state,particularly at the early stage of the evolution.It is worth noting that the optical coupling sets up a correlation between the two cavity modes,which has significant effect on the cooling process.Using numerical solutions,we find that the inter-cavity coupling will decrease the cooling effect when both cavities have the same effective optomechanical coupling.However,when the effective optomechanical couplings are different,the cooling effect will be strongly improved by selecting appropriate range of inter-cavity coupling.