Find the number of solutions, in positive real numbers, to the system
\(\begin{align*} x_1 + \frac{1}{x_2} &= 4, \\ x_2 + \frac{1}{x_3} &= 1, \\ x_3 + \frac{1}{x_4} &= 4, \\ x_4 + \frac{1}{x_5} &= 1, \\ &\dots, \\ x_{99} + \frac{1}{x_{100}} &= 4, \\ x_{100} + \frac{1}{x_1} &= 1. \end{align*}\)
