A770.Cow Gymnasts--Platinum

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, $N$ ($1 \leq N \leq 10^{12}$).

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 $(3,2,3,2)$.