A rational function $f(x) = \frac{ax^2 + bx + c}{dx^2 + ex + f}$ has the following properties: a removable discontinuity at $x = 2$, a vertical asymptote at $x = -3$, a horizontal asymptote at $y = 1$, and $f(0) = -\frac{2}{3}$. Determine the sum $a + b + c + d + e + f$.