I am trying to find information about the following problem. Let $C_1$ and $C_2$ be any two poly-size arithmetic circuits on input vectors x₁ and x₂ correspondingly. Assume that a third arithmetic circuit $C_3$ (also poly-size) combines their outputs. To make the problem non-trivial, $C_3$ can have multiplicative-depth $1$; however, the other two circuits have no such restrictions. What is known about the power of the resulting circuit class? Even some good keywords would help. Thanks in advance.
edit: forgot some details, please see my comments.