Find The Largest N

Consider the function $f(x) = \dfrac{(x-1)^{1093^{2014}} - x^{1093^{2014}} + 1}{1093(x-1)}.$ Find the last three digits of the largest positive integer $N$ such that there exist two polynomials $a(x)$ and $b(x)$ both having integer coefficients and an integer $c$ (not necessarily positive) satisfying $f(x) = (x-c)^N a(x) + b(x).$

Details and assumptions

• You might use the fact that $1093$ is a prime and $2^{1092}-1$ is a multiple of $1093^2$ (such primes are called Wieferich primes).

• This problem is not original.