Let \(f\) be a function from the positive integers to the positive integers such that \[f(m+n) + 2 > f(m) + f(f(n))\] for all positive integers \(m,n.\) Find the last three digits of the sum of all possible values of \(f(2014).\)

**Details and assumptions**

- The inequality is strict.
- This problem is not original.
- A typo has been fixed. Sorry for the inconvenience.

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