Level pending

Let $$a$$ and $$b$$ be positive integers such that $$a^2+b^2 \equiv 2014 \pmod{a+b}$$ and $$a^3 + b^3\equiv 2014^2 \pmod{a^2+b^2}$$. Evaluate the digit sum of

$$\dfrac{a^3+b^3}{a+b}$$.

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