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For fixed integers a, b, c, define the crazy function F(n) as follows:
F(n) = n - c for all n > b
F(n) = F(a + F(a + F(a + F(a + n)))) for all n ≤ b.
Also, define $S(a, b, c) = \sum \limits_{n = 0}^{b} {F(n)}$.
For example, if a = 50, b = 2000 and c = 40, then F(0) = 3240 and F(2000) = 2040.
Also, S(50, 2000, 40) = 5204240.
Find the last 9 digits of S(217, 721, 127).
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