A researcher is analyzing a dataset of $n$ observations. They discover that the sum of the squared deviations from the mean is exactly equal to the square of the sample size, $\sum_{i=1}^{n} (x_i - \bar{x})^2 = n^2$. If the researcher decides to increase every data point by a constant $k$ and then scale the entire dataset by a factor $c > 0$, what is the expression for the new sample variance $s^2_{new}$ in terms of $n$ and $c$?