Putnam 2001 B2

Find all pairs of real numbers $(x,y)$ satisfying the system of equations

$$\begin{aligned}\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2) \\ \frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4).\end{aligned}$$