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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