  Non-reciprocal lattice systems are among the simplest non-Hermitian systems,
exhibiting several key features absent in their Hermitian counterparts. In this
study, we investigate the Hatano-Nelson model with an impurity and unveil how
the impurity influences the intrinsic non-Hermitian skin effect of the system.
We present an exact analytical solution to the problem under both open and
periodic boundary conditions, irrespective of the impurity's position and
strength. This exact solution is thoroughly validated by numerical simulations.
Our analysis reveals a distinctive phenomenon where a specific impurity
strength, determined by the non-reciprocal hopping parameters, induces a unique
skin state at the impurity site. This impurity state exhibits a skin effect
that counterbalances the boundary-induced skin effect, a phenomenon we term the
impurity-induced counter skin-effect. These findings offer unprecedented
insights into the dynamics of non-Hermitian systems with impurities,
elucidating the complex interplay between impurities and the system's
non-reciprocal nature. Furthermore, our results suggest potential applications
in quantum sensing, where the unique characteristics of the impurity-induced
counter skin-effect could be harnessed for enhanced sensitivity and precision.
