Modulo power identity

Let S(n) be the sum of all positive integers m not exceeding n having the following property:a ^m+4 ≡ a (mod m) for all integers a.

The values of m ≤ 100 that satisfy this property are 1, 2, 3, 5 and 21, thus S(100) = 1+2+3+5+21 = 32. You are given S(10⁶) = 22868117.

Find S(10¹²).

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