A dataset consists of $n$ distinct positive numbers with mean $\bar{x}$. Removing the largest element, $x_{\max}$, reduces the mean of the remaining $n-1$ elements to $\bar{x} - d$. Removing instead the smallest element, $x_{\min}$, increases the mean of the remaining $n-1$ elements to $\bar{x} + c$. Find the range of the original dataset ($x_{\max} - x_{\min}$) in terms of $n, c, \text{ and } d$.