USACO 2018 February Contest, Platinum

Problem 3. Cow Gymnasts

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Bored of farm life, the cows have sold all their earthly possessions and joined the crew of a traveling circus. So far, the cows had been given easy acts: juggling torches, walking tightropes, riding unicycles -- nothing a handy-hoofed cow couldn't handle. However, the ringmaster wants to create a much more dramatic act for their next show.

The stage layout for the new act involves $N$ platforms arranged in a circle. On each platform, between $1$ and $N$ cows must form a stack, cow upon cow upon cow. When the ringmaster gives the signal, all stacks must simultaneously fall clockwise, so that the bottom cow in a stack doesn't move, the cow above her moves one platform clockwise, the next cow moves two platforms clockwise, and so forth. Being accomplished gymnasts, the cows know they will have no trouble with the technical aspect of this act. The various stacks of cows will not "interfere" with each other as they fall, so every cow will land on the intended platform. All of the cows landing on a platform form a new stack, which does not fall over.

The ringmaster thinks the act will be particularly dramatic if after the stacks fall, the new stack on each platform contains the same number of cows as the original stack on that platform. We call a configuration of stack sizes "magical" if it satisfies this condition. Please help the cows by computing the number of magical configurations. Since this number may be very large, compute its remainder modulo $10^9 + 7$.

Two configurations are considered distinct if there is any platform for which the configurations assign a different number of cows.


The input is a single integer, $N$ ($1 \leq N \leq 10^{12}$).

OUTPUT FORMAT (file gymnasts.out):

A single integer giving the number of magical configurations modulo $10^9 + 7$.




For $N = 4$, the valid configurations are $(1,1,1,1)$, $(2,2,2,2)$, $(3,3,3,3)$, $(4,4,4,4)$, $(2,3,2,3)$, and $(3,2,3,2)$.

Problem credits: Dhruv Rohatgi

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