Thanks to the availability of exact results, two dimensional spin systems are often taken as a reference frame to test novel Monte Carlo techniques. In this context, the Potts model at large number of states can be used to assess the ability of a numerical method to overcome strong metastabilities. In this talk, I will discuss a recently introduced Monte Carlo method (known in the literature as the LLR method) that has been proved to be very efficient at dealing with metastabilities arising at first order transition points and I will compare numerical results with the exact results on a q=20 Potts model. The striking outcome of the study is that the numerical determinations of thermodynamic observables (including the order-disorder interface tension at criticality) displays an impressive degree of accuracy with respect to the exact value (down to a level of a few parts in a million). This confirms the reliability and the efficiency of the LLR method, suggesting its applicability also in scenarios in which exact results are not available. Indeed I will show that the method enables us to go well beyond the state of the art in a four-dimensional compact U(1) model at criticality.