MAGIC is a system of two Imaging Atmospheric Cherenkov Telescopes (IACTs), designed to detect very-high-energy gamma rays, and is operating in stereoscopic mode since 2009 at the Observatorio del Roque de Los Muchachos in La Palma, Spain. In 2018, the prototype IACT of the Large-Sized Telescope (LST-1) for the Cherenkov Telescope Array, a next-generation ground-based gamma-ray observatory, was inaugurated at the same site, at a distance of approximately 100 meters from the MAGIC telescopes. Using joint observations between MAGIC and LST-1, we developed a dedicated analysis pipeline and established the threefold telescope system via software, achieving the highest sensitivity in the northern hemisphere. Based on this enhanced performance, MAGIC and LST-1 have been jointly and regularly observing the Galactic Center, a region of paramount importance and complexity for IACTs. In particular, the gamma-ray emission from the dynamical center of the Milky Way is under debate. Although previous measurements suggested that a supermassive black hole Sagittarius A* plays a primary role, its radiation mechanism remains unclear, mainly due to limited angular resolution and sensitivity. The enhanced sensitivity in our novel approach is thus expected to provide new insights into the question. We here present the current status of the data analysis for the Galactic Center joint MAGIC and LST-1 observations.