Determine the values of the positive integer $n$ for which the following system of equations has a solution in positive integers $x_1, x_2,...,, x_n$. Find all solutions for each such $n$. $$\begin{cases} x_1 + x_2 +...+ x_n = 16 \\ \\ \dfrac{1}{x_1} + \dfrac{1}{x_2} +...+ \dfrac{1}{x_n} = 1\end{cases}$$