A convex polyhedron $Q$ has vertices $V_1,V_2,\ldots,V_n$, and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\ldots,P_n$ 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 edges does $R$ have? $\textbf{(A)}\ 200\qquad$ $ \textbf{(B)}\ 2n\qquad$ $ \textbf{(C)}\ 300\qquad$ $ \textbf{(D)}\ 400\qquad$ $ \textbf{(E)}\ 4n$