A convex polyhedron $Q$ has vertices $V_1,V_2,\cdots,V_n$, and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\cdots,P_n$ 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?

Let's think step by step and output the final answer within \boxed{}.
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