Negative pairs yield symmetric solutions. Total lattice points: $(\pm26, \pm12)$ and $(\pm10, 0)$, totaling $\boxed6$.Question: A mechanical engineer is designing a gear system where the torque output $ T(n) $ at segment $ n $ is defined recursively by $ T(n) = 2T(n-1) + 3 $, with $ T(1) = 4 $. If the system evaluates $ T(m) $ for a specific $ m $, and the sum $ S(m) = \sum_n=1^m T(n) $ models total energy over time, compute $ S(5) $. - Abu Waleed Tea