Numbers on a Blackboard, Extreme Edition

There are 2015 one's on a blackboard. Every turn, I erase two numbers $$a$$ and $$b$$ on the blackboard and replace them with $\dfrac{4ab-1}{4(a+b-1)}$

After 2014 turns, there is one number remaining. Find the number of possible values of this number.

Details and Assumptions:

If the number you get is over 1000, then submit your answer as the last three digits of that number.

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