KöMaL 2025 A896

Marine biologists are studying a new species of shellfish whose first generation consists of 100 shellfish, and their colony reproduces as follows: if a given generation consists of $N$ shellfish (where $5 \mid N$ always holds), they divide themselves into $N/5$ groups of 5 shellfish each. Each group collectively produces 15 offspring, who form the next generation. Some of the shellfish contain a pearl, but a shellfish can only contain a pearl if none of its direct ancestors contained a pearl. The value of a pearl is determined by the generation of the shellfish containing it: in the $n$-th generation, its value is $1 / 3^n$. Find the maximum possible total value of the pearls in the colony.