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Problem 202

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Laserbeam

Three mirrors are arranged in the shape of an equilateral triangle, with their reflective surfaces pointing inwards. There is an infinitesimal gap at each vertex of the triangle through which a laser beam may pass.

Label the vertices A, B and C. There are 2 ways in which a laser beam may enter vertex C, bounce off 11 surfaces, then exit through the same vertex: one way is shown below; the other is the reverse of that.

There are 80840 ways in which a laser beam may enter vertex C, bounce off 1000001 surfaces, then exit through the same vertex.

In how many ways can a laser beam enter at vertex C, bounce off 12017639147 surfaces, then exit through the same vertex?

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激光束

三面镜子摆成了正三角形的形状，其反射面朝内放置。在三角形的顶点处有一个无穷小的间隙可容一束激光束通过。

记三个顶点分别为A、B和C。从顶点C射入的激光束，经过11次反射后再从同一个顶点射出，这样的路径一共有两条：其中一条如下所示，另一条即是将其反向。

从顶点C射入的激光束，经过1000001次反射后再从同一个顶点射出，这样的路径一共有80840条。

从顶点C射入的激光束，经过12017639147次反射后再从同一个顶点射出，这样的路径一共有多少条？

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