Let $(Z_i)_{i=1}^{\infty}$ be an i.i.d sequence of random variables such that $\mathbb{P}(Z_i=1) = p$, $\mathbb{P}(Z_i=0) = 1-p$ and $(V_i)_{i=1}^{\infty}$ be an i.i.d sequence of random variables such that $\mathbb{P}(V_i=1)=q$, $\mathbb{P}(V_i)=0)=1-q$. For all $i,j$ variables $Z_i$ and $V_i$ are independent. Let $\kappa$ be an Poisson distributed random variable independent of all $V_i$ and $Z_i$. Find the distribution of $\sum_{i=1}^{\kappa} (V_i + Z_i)^2$.
I know that if $A_i$ have Bernoulli distribution, i.e. $\mathbb{P}(A_i=1)=p$, $\mathbb{P}(A_i = 0) = 1-p$, and $\kappa$ is independent of all $A_i$-s and of $Poiss(\lambda)$ distribution, then $\sum_{i=1}^{\kappa} (V_i + Z_i)^2$ has $Poiss(p \lambda)$ distribution. But $(V_i+Z_i)^2$ can take three values (0, 1 and 4), not two, so I'm not really sure how to do this.
Edit: obtaining Laplace transform of this sum is also interesting for me.