Two sequences $b_i$, $c_i$, $0 \le i \le 100$ contain positive integers, except $c_0=0$ and $b_{100}=0$. Some towns in Graphland are connected with roads, and each road connects exactly two towns and is precisely $1$ km long. Towns, which are connected by a road or a sequence of roads, are called neighbours. The length of the shortest path between two towns $X$ and $Y$ is denoted as distance. It is known that the greatest distance between two towns in Graphland is $100$ km. Also the following property holds for every pair $X$ and $Y$ of towns (not necessarily distinct): if the distance between $X$ and $Y$ is exactly $k$ km, then $Y$ has exactly $b_k$ neighbours that are at the distance $k+1$ from $X$, and exactly $c_k$ neighbours that are at the distance $k-1$ from $X$. Prove that $$\frac{b_0b_1 \cdot \cdot \cdot b_{99}}{c_1c_2 \cdot \cdot \cdot c_{100}}$$is a positive integer.