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In this talk, I will discuss a new spinorial strategy for constructing geometric inequalities involving the ADM mass of initial data for vacuum spacetime with a black hole. This approach is based on a second-order elliptic equation for a valence 1 Weyl spinor. It offers an advantage over other spinorial approaches to constructing geometric inequalities based on the Sen-Witten-Dirac equation, as it allows for specifying boundary conditions for the two components of the spinor. This greater control over the boundary data has the potential to give rise to new geometric inequalities involving the mass. In particular, I will show that the mass is bounded from below by an integral functional over a marginally outer trapped surface (MOTS) that depends on a freely specifiable valence 1 spinor. From this main inequality, by choosing the free data appropriately, one obtains new nontrivial bounds of the mass in terms of the inner expansion of the MOTS and its quasilocal angular momentum. The analysis makes use of a new formalism for the 1+1+2 decomposition of spinorial equations. This work was done in collaboration with J. Valiente Kroon (London) and A. Soria (Madrid).