Recall that the conjugate of the complex number $w = a + bi$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$, is the complex number $\overline{w} = a - bi$. For any complex number $z$, let $f(z) = 4i\overline{z}$. The polynomial  
$$P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1$$  
has four complex roots: $z_1$, $z_2$, $z_3$, and $z_4$. Let  
$$Q(z) = z^4 + Az^3 + Bz^2 + Cz + D$$ 
be the polynomial whose roots are $f(z_1)$, $f(z_2)$, $f(z_3)$, and $f(z_4)$, where the coefficients $A,$ $B,$ $C,$ and $D$ are complex numbers. Find the value of $B + D$.

Let's think step by step and output the final answer within \boxed{}.
user