The active-space coupled-cluster (CC) and equation-of-motion (EOM) CC methods with all single and double excitations with triple excitations defined via active orbitals (CCSDt, EOMCCSDt), as implemented with TENSOR CONTRACTION ENGINE, are applied to the challenging Be3 system, which is characterized by a large number of low-lying excited states dominated by two-electron transitions and significant high-order correlation effects in the ground electronic state. It is demonstrated that the CCSDt and EOMCCSDt methods provide an excellent description of complicated electronic quasidegeneracies present in the Be3 cluster. Different strategies for defining triple excitations within the CCSDtEOMCCSDt approach are discussed.