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The heavy ion (HI) program at the LHC has proven to be a successful and indispensable part of the LHC physics program. Its chief aim had been the detailed characterization of the quark-gluon plasma (QGP) in lead-lead collisions. Using additional data sets of proton-lead, proton-proton, and xenon-xenon collisions, the program has also included many advances, for example, in the understanding of the partonic nuclear structure, collectivity in smaller collision systems, and electromagnetic interactions. This Letter of Interest outlines the CMS Heavy Ion Group point of view regarding the scientific case for the use of ultrarelativistic HI beams in the coming decade to characterize QGP with unparalleled precision and to probe novel fundamental physics phenomena. More specifically, it outlines the open questions in the field which can be addressed with CMS, and aims to promote engagement from the US community and its international partners by building upon the recently concluded Snowmass 2022 exercise, the input provided to the European Strategy for Particle Physics, and proposed continuations and extensions of the last version of the US Long-Range Plan for Nuclear Physics.