Given an elliptic curve $y^2 = x^3 + ax + b$ with $4a^3 + 27b^2 \neq 0$ over $R$, derive the formulas of group operations. That is, given distinct $(x_0, y_0)$ and $(x_1, y_1)$ on the curve, derive the formulas of addition $(x_2, y_2) = (x_0, y_0) + (x_1, y_1)$ and doubling $(x_3, y_3) = 2(x_0, y_0)$.