A convex polyhedron $Q$ has vertices $V_1,V_2,\dots ,V_n,$ and $120$ edges. The polyhedron is cut by planes $P_1,P_2,\dots ,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. How many vertices does $R$ have?