Existence of a complex polynomial with given zeros and critical points?

Question

Suppose we are given two families of complex numbers (need not to be distinct) $z_1, z_2, \cdots, z_n$ and $w_1, w_2, \cdots, w_{n-1}$ such that

1. the second family lies in the convex hull of first family and
2. both families have the same center of mass.

   Now, can we find a polynomial $P$ of degree $n$ that has first family as its collection of zeros and second family as its collection of critical points (counting with multiplicities) ?

Hermite interpolation gives a polynomial of degree $2n-1$ with these properties introducing new zeros and critical points, but I hope conditions 1. and 2. are enough to guarantee the existence of such a polynomial of degree $n.$ If not, what additional conditions do we need to impose of those two families?