This symmetric utilization of the two states (M s = ±1) of the triplet allows us to address the problematic spin contamination of SF-TDDFT 56 through a formulational structure, which we call "external contraction." By pre-defining the equations of motion for each spin state, we eliminate the need to identify them individually. As MRSF utilizes the triplet reference state, both the ground singlet and all excited singlet states can be obtained on an equal footing. This approach not only eliminates the general topological problem of conical intersections present in all single-reference theories but also allows for the description of open-shell ground singlet states, such as diradicals and Jahn-Teller distortions, as well as bond-breaking events. The current implementation of MRSF-TDDFT is based on a restricted open-shell Kohn-Sham (ROKS) formalism, chosen for its simplicity and freedom from spin contamination and orbital asymmetries between α and β orbitals. In contrast, the unrestricted Kohn-Sham (UKS) approach offers the advantage of variational flexibility, as both α and β orbitals are independently optimized. Generally, MRSF-TDDFT based on UKS is expected to be more adaptable in various strongly correlated situations compared to ROKS formulations. The development of this approach is the main focus of the current paper.