MathNet 2015-part-03 — 191 problems (EN–VI) Question 165

When five positive integers $a, b, c, d, e$ satisfy $a \lt b \lt c \lt d \lt e \lt a^2 \lt b^2 \lt c^2 \lt d^2 \lt e^2 \lt a^3 \lt b^3 \lt c^3 \lt d^3 \lt e^3$, determine the minimum possible value that the sum $a+b+c+d+e$ can take.