$Kepler$ revealed that roughly one-third of Sun-like stars host planets orbiting within 100 days and between the size of Earth and Neptune. How do these planets form, what are they made of, and do they represent a continuous population or multiple populations? To help address these questions, we began the Magellan-TESS Survey (MTS), which uses Magellan II/PFS to obtain radial velocity (RV) masses of 30 TESS-detected exoplanets and develops an analysis framework that connects observed planet distributions to underlying populations. In the past, small planet RV measurements have been challenging to obtain due to host star faintness and low RV semi-amplitudes, and challenging to interpret due to the potential biases in target selection and observation planning decisions. The MTS attempts to minimize these biases by focusing on bright TESS targets and employing a quantitative selection function and observing strategy. In this paper, we (1) describe our motivation and survey strategy, (2) present our first catalog of planet density constraints for 27 TESS Objects of Interest (TOIs; 22 in our population analysis sample, 12 that are members of the same systems), and (3) employ a hierarchical Bayesian model to produce preliminary constraints on the mass-radius (M-R) relation. We find that the biases causing previous M-R relations to predict fairly high masses at $1~R_oplus$ have been reduced. This work can inform more detailed studies of individual systems and offer a framework that can be applied to future RV surveys with the goal of population inferences.
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