A force field is given by $\vec{F}(x, y) = \langle 2xy^3 + 3x^2 y, 3x^2 y^2 + x^3 + 1 \rangle$. A particle of mass $m = 2$ moves from $(0, 0)$ with an initial speed of $v_0 = 3$ to $(1, 2)$ under the influence of this force field and no other forces. Find the final speed $v_f$ of the particle.