The alternating harmonic series $S = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ converges conditionally to $\ln 2$. Consider the rearrangement $S' = 1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - \dots$ where one positive term is followed by two negative terms. What is the sum of $S'$?