Determining the existence and the location of the QCD critical point remains a major open problem, both theoretically and experimentally. In this talk, I present a new way of reconstructing the equation of state in the vicinity of the nearest thermodynamic singularity (the Lee-Yang edge singularity in the crossover region) from a truncated Taylor series expansion for small μ. This is done by using a combination of Padé resummation and a conformal map. Then, I show that this information can be used to (i) determine the location of the critical point and (ii) constrain the non-universal mapping parameters between the Ising and QCD equations of state. I explicitly demonstrate these ideas in the 2d Gross-Neveu model whose phase diagram shares the key aspects of the conjectured QCD phase diagram including the existence of a critical point.