Modular Madness

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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